RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2

RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2

Other Exercises

Question 1.
Which among the following are nets for a cube ?
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 1
Solution:
Nets for a cube are (ii), (iv) and (vi)

Question 2.
Name the polyhedron that can be made by folding each net:
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 2
Solution:
(i) This net is for a square
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 3
(ii) This net is for triangular prism.
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 4
(iii) This net is for triangular prism.
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 5
(iv) This net is for hexagonal prism.
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 6
(v) This net is for hexagon pyramid.
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 7
(vi) This net is for cuboid.
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 8

Question 3.
Dice are cubes where the numbers on the opposite faces must total 7. Which of the following are dice ?
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 9
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 10
Solution:
Figure (i) shows the net of cube or dice.

Question 4.
Draw nets for each of the following polyhedrons:
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 11
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 12
Solution:
(i) Net for cube is given below :
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 13
(ii) Net of a triangular prism is as under :
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 14
(iii) Net of hexagonal prism is as under :
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 15
(iv) The net for pentagonal pyramid is as under:
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 16

Question 5.
Match the following figures:
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 17
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.2 18
Solution:
(a) (iv)
(b) (i)
(c) (ii)
(d) (iii)

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RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.1

RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.1

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.1

Other Exercises

Question 1.
What is the least number of planes that can enclose a solid ? What is the name of the solid ?
Solution:
The least number of planes that can enclose a solid is called a Tetrahedron.

Question 2.
Can a polyhedron have for its faces :
(i) three triangles ?
(ii) four triangles ?
(iii) a square and four triangles ?
Solution:
(i) No, polyhedron has three faces.
(ii) Yes, tetrahedron has four triangles as its faces.
(iii) Yes, a square pyramid has a square as its base and four triangles as its faces.

Question 3.
Is it possible to have a polyhedron with any given number of faces ?
Solution:
Yes, it is possible if the number of faces is 4 or more.

Question 4.
Is a square prism same as a cube ?
Solution:
Yes, a square prism is a cube.

Question 5.
Can a polyhedron have 10 faces, 20 edges and 15 vertices ?
Solution:
No, it is not possible as By Euler’s formula
F + V = E + 2
⇒ 10 + 15 = 20 + 2
⇒ 25 = 22
Which is not possible

Question 6.
Verify Euler’s formula for each of the following polyhedrons :
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.1 1
Solution:
(i) In this polyhedron,
Number of faces (F) = 7
Number of edges (E) = 15
Number of vertices (V) = 10
According to Euler’s formula,
F + V = E + 2
⇒ 7 + 10 = 15 + 2
⇒ 17 = 17
Which is true.
(ii) In this polyhedron,
Number of faces (F) = 9
Number of edges (E) = 16
Number of vertices (V) = 9
According to Euler’s formula,
F + V = E + 2
⇒ 9 + 9 = 16 + 2
⇒ 18 = 18
Which is true.
(iii) In this polyhedron,
Number of faces (F) = 9
Number of edges (E) =18
Number of vertices (V) = 11
According to Euler’s formula,
F + V = E + 2
⇒ 9 + 11 = 18 + 2
⇒ 20 = 20
Which is true.
(iv) In this polyhedron,
Number of faces (F) = 5
Number of edges (E) = 8
Number of vertices (V) = 5
According to Euler’s formula,
F + V = E + 2
⇒ 5 + 5 = 8 + 2
⇒ 10 = 10
Which is true.
(v) In the given polyhedron,
Number of faces (F) = 9
Number of edges (E) = 16
Number of vertices (V) = 9
According to Euler’s formula,
F + V = E + 2
⇒ 9 + 9 = 16 + 2
⇒ 18 = 18
Which is true.

Question 7.
Using Euler’s formula, find the unknown:
RD Sharma Class 8 Solutions Chapter 19 Visualising Shapes Ex 19.1 2
Solution:
We know that Euler’s formula is
F + V = E + 2
(i) F + 6 = 12 + 2
⇒ F + 6 = 14
⇒ F = 14 – 6 = 8
Faces = 8
(ii) F + V = E + 2
⇒ 5 + V = 9 + 2
⇒ 5 + V = 11
⇒ V = 11 – 5 = 6
Vertices = 6
(iii) F + V = E + 2
⇒ 20 + 12 = E + 2
⇒ 32 = E + 2
⇒ E = 32 – 2 = 30
Edges = 30

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RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.5

RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.5

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.5

Other Exercises

Question 1.
Construct a quadrilateral ABCD given that AB = 4 cm, BC = 3 cm, ∠A = 75°, ∠B = 80° and ∠C = 120°.
Solution:
Steps of construction :
(i) Draw a line segment AB = 4 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.5 1
(ii) At A draw a ray AX making an angle of 75°.
(iii) At B draw another ray BY making an angle of 80° and cut off BC = 3 cm.
(iv) At C, draw another ray CZ making an angle of 120° which intersects AX at D.
Then ABCD is the required quadrilateral.

Question 2.
Construct a quadrilateral ABCD where AB = 5.5 cm, BC = 3.7 cm, ∠A = 60°, ∠B = 105° and ∠D = 90°.
Solution:
∠A = 60°, ∠B = 105° and ∠D = 90°
But ∠A + ∠B + ∠C + ∠D = 360° (Sum of angles of a quadrilateral)
⇒ 60° + 105° + ∠C + 90° = 360°
⇒ 255° + ∠C = 360°
⇒ ∠C = 360° – 255° = 105°
Steps of construction :
(i) Draw a line segment AB = 5.5 cm.
(ii) At A, draw a ray AX making an angle of
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.5 2
(iii) At B, draw another ray BY making an angle of 105° and cut off BC = 3.7 cm.
(iv) At C, draw a ray CZ making an angle of 105° which intersects AX at D.
Then ABCD is the required quadrilateral.

Question 3.
Construct a quadrilateral PQRS where PQ = 3.5 cm, QR = 6.5 cm, ∠P = ∠R = 105° and ∠S = 75°.
Solution:
∠P = 105°, ∠R = 105° and ∠S = 75°
But ∠P + ∠Q + ∠R + ∠S = 360° (Sum of angles of a quadrilateral)
⇒ 105° + ∠Q + 105° + 75° = 360°
⇒ 285° + ∠Q = 360°
⇒ ∠Q = 360° – 285° = 75°
Steps of construction :
(i) Draw a line segment PQ = 3.5 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.5 3
(ii) At P, draw a ray PX making an angle of 105°.
(iii) At Q, draw another ray QY, making an angle of 75° and cut off QR = 6.5 cm.
(iv) At R, draw a ray RZ making an angle of 105° which intersects PX at S.
Then PQRS is the required quadrilateral.

Question 4.
Construct a quadrilateral ABCD when BC = 5.5 cm, CD = 4.1 cm, ∠A = 70°, ∠B = 110° and ∠D = 85°.
Solution:
∠A = 70°, ∠B = 110°, ∠D = 85°
But ∠A + ∠B + ∠C + ∠D = 360° (Sum of angles of a quadrilateral)
⇒ 70° + 110° + ∠C + 85° = 360°
⇒ 265° + ∠C = 360°
⇒ ∠C = 360° – 265° = 95°
Steps of construction:
(i) Draw a line segment BC = 5.5 cm.
(ii) At B, draw a ray BX making an angle of 110°.
(iii) At C, draw another ray CY making an angle of 95° and cut off CD = 4.1 cm.
(iv) At D, draw a ray DZ making an angle of 85° which intersects BX at A.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.5 4
Then ABCD is the required quadrilateral.

Question 5.
Construct a quadrilateral ABCD, where ∠A = 65°, ∠B = 105°, ∠C = 75°, BC = 5.7 cm and CD = 6.8 cm.
Solution:
∠A = 65°, ∠B = 105°, ∠C = 75°
But ∠A + ∠B + ∠C + ∠D = 360° (Sum of angles of a quadrilateral)
⇒ 65° + 105° + 75° + ∠D = 360°
⇒ 245° + ∠D = 360°
⇒ ∠D = 360° – 245° = 115°
Steps of construction:
(i) Draw a line segment BC = 5.7 cm.
(ii) At B, draw a ray BX making an angle of
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.5 5
(iii) At C draw a another ray CY making an angle of 75° and cut off CD = 6.8 cm.
(iv) At D, draw a ray DZ making an angle of 115° which intersects BX at A.
Then ABCD is the required quadrilateral.

Question 6.
Construct a quadrilateral PQRS in which PQ = 4 cm, QR = 5 cm, ∠P = 50°, ∠Q = 110° and ∠R = 70°.
Solution:
Steps of construction :
(i) Draw a line segment PQ = 4 cm.
(ii) At P, draw a ray PX making an angle of 50°.
(iii) At Q, draw another ray QY making an angle of 110° and cut off QR = 5 cm.
(iv) At R, draw a ray RZ making an angle of 70° which intersects PX at S.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.5 6
Then PQRS is the required quadrilateral.

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RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4

RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4

Other Exercises

Question 1.
Construct a quadrilateral ABCD, in which AB = 6 cm, BC = 4 cm, CD = 4 cm, ZB = 95° and ∠C = 90°.
Solution:
Steps of construction :
(i) Draw a line segment BC = 4 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4 1
(ii) At B, draw a ray BX making an angle of 95° and cut off BA = 6 cm.
(iii) At C, draw a ray CY making an angle of 90° and cut off CD = 4 cm.
(iv) Join AD.
Then ABCD is the required quadrilateral.

Question 2.
Construct a quadrilateral ABCD, where AB = 4.2 cm, BC = 3.6 cm, CD = 4.8 cm, ∠B = 30° and ∠C = 150°.
Solution:
Steps of construction :
(i) Draw a line segment BC = 3.6 cm.
(ii) At B, draw a ray BX making an angle of 30° and cut of BA = 4.2 cm.
(iii) At C, draw another ray CY making an angle of 150° and cut off CD = 4.8 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4 2
(iv) Join AD.
Then ABCD is the required quadrilateral.

Question 3.
Construct a quadrilateral PQRS, in which PQ = 3.5 cm, QR = 2.5 cm, RS = 4.1 cm, ∠Q = 75° and ∠R = 120°.
Solution:
Steps of construction :
(i) Draw a line segment QR = 2.5 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4 3
(ii) At Q, draw a ray QX making an angle of 75° and cut off QP = 3.5 cm.
(iii) At R, draw another ray RY making an angle of 120° and cut off RS = 4.1 cm.
(iv) Join PS.
Then PQRS is the required quadrilateral.

Question 4.
Construct a quadrilateral ABCD given BC = 6.6 cm, CD = 4.4 cm, AD = 5.6 cm and ∠D = 100° and ∠C = 95°.
Solution:
Steps of construction :
(i) Draw a line segment CD = 4.4 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4 4
(ii) At C, draw a ray CX making an angle of 95° and cut off CB = 6.6 cm
(iii) At D, draw another ray DY making an angle of 100° and cut off DA = 5.6 cm.
(iv) Join AB.
Then ABCD is the required quadrilateral.

Question 5.
Construct a quadrilateral ABCD in which AD = 3.5 cm, AB = 4.4 cm, BC = 4.7 cm, ∠A = 125° and ∠B = 120°.
Solution:
Steps of construction :
(i) Draw a line segment AB 4.4 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4 5
(ii) At A, draw a ray AX making an angle of 125° and cut off AD = 3.5 cm.
(iii) At B, draw another ray BY making an angle of 120° and cut off BC = 4.7 cm.
(iv) Join CD.
Then ABCD is the required quadrilateral.

Question 6.
Construct a quadrilateral PQRS in which ∠Q = 45°, ∠R = 90°, QR = 5 cm, PQ = 9 cm and RS = 7 cm.
Solution:
Steps of construction :
This quadrilateral is not possible to construct as shown in the figure.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4 6

Question 7.
Construct a quadrilateral ABCD in which AB = BC = 3 cm, AD = 5 cm, ∠A = 90° and ∠B = 105°.
Solution:
Steps of construction :
(i) Draw a line segment AB = 3 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4 7
(ii) At A, draw a ray AX making an angle of 90° and cut off AD = 5 cm.
(iii) At B, draw another ray BY making an angle of 105° and cut off BC = 3 cm.
(iv) Join CD.
Then ABCD is the required quadrilateral.

Question 8.
Construct a quadrilateral BDEF where DE = 4.5 cm, EF = 3.5 cm, FB = 6.5 cm and ∠F = 50° and ∠E = 100°
Solution:
Steps of construction :
(i) Draw a line segment EF = 3.5 cm.
(ii) At E, draw a ray EX making an angle of 100° and cut off ED = 4.5 cm.
(iii) At F, draw another ray FY making an angle of 45° and cut off FB = 6.5 cm.
(iv) Join DB.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.4 8
Then BDEF is the required quadrilateral.

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RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.3

RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.3

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.3

Other Exercises

Question 1.
Construct a quadrilateral ABCD in which AB = 3.8 cm, BC = 3.4 cm, CD = 4.5 cm, AD = 5 cm and ∠B = 80°.
Solution:
Steps of construction :
(i) Draw a line segment AB = 3.8 cm.
(ii) At B, draw a ray BX making an angle of 80° and cut off BC = 3.4 cm.
(iii) With centre A and radius 5 cm and with centre C and radius 4.5 cm, draw arcs which intersect each other at D.
(iv) Join CD and AD.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.3 1
ABCD is the required quadrilateral.

Question 2.
Construct a quadrilateral ABCD given that AB = 8 cm, BC = 8 cm, CD = 10 cm, AD = 10 cm and ∠A = 45°.
Solution:
Steps of construction :
(i) Draw a line segment AB = 8 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.3 2
(ii) At A, draw a ray AX making an angle of 45° and cut off BC = 8 cm.
(iii) With centre A and C and radius 10 cm, draw arcs intersecting each other at D.
(iv) Join AD, CD.
Then ABCD is the required quadrilateral.

Question 3.
Construct a quadrilateral ABCD in which AB = 7.7 cm, BC = 6.8 cm, CD = 5.1 cm, AD = 3.6 cm and ∠C = 120°.
Solution:
Steps of construction :
(i) Draw a line segment BC = 6.8 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.3 3
(ii) At C, draw a ray CX making an angle of 120° and cut off CD = 5.1 cm.
(iii) With centre B and radius 7.7 cm and with centre D and radius 3.6 cm draw arcs which intersect each other at A.
(iv) Join AD and AB.
Then ABCD is the required quadrilateral.

Question 4.
Construct a quadrilateral ABCD in which AB = BC = 3 cm, AD = CD = 5 cm and ∠B = 120°
Solution:
Steps of construction :
(i) Draw a line segment AB = 3 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.3 4
(ii) At B, draw a ray BX making an angle of 120° and cut off BC = 3 cm.
(iii) With centres A and C, and radius 5 cm, draw arcs intersecting each other at D.
(iv) Join AD and CD.
Then ABCD is the required quadrilateral.

Question 5.
Construct a quadrilateral ABCD in which AB = 2.8 cm, BC = 3.1 cm, CD = 2.6 cm and DA = 3.3 cm and ∠A = 60°.
Solution:
Steps of construction :
(i) Draw a line segment AB = 2.8 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.3 5
(ii) At A draw a ray AX making an angle of 60° and cut off AD = 3.3 cm.
(iii) With centre B and radius 3.1 cm and with centre D and radius 2.6 cm, draw arc which intersect each other at C.
(iv) Join CB and CD.
Then ABCD is the required quadrilateral.

Question 6.
Construct a quadrilateral ABCD in which AB = BC = 6 cm, AD = DC = 4.5 cm and ∠B = 120°.
Solution:
Steps of construction:
The construction is not possible to draw as arcs of radius 4.5 cm from A and C, do not intersect at any point.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.3 6

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RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.2

RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.2

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.2

Other Exercises

Question 1.
Construct a quadrilateral ABCD in which AB = 3.8 cm, BC = 3.0 cm, AD = 2.3 cm, AC = 4.5 cm and BD = 3.8 cm.
Solution:
Steps of construction :
(i) Draw a line segment AB = 3.8 cm.
(ii) With centre A and radius 2.3 cm and with centre B and radius 3.8 cm draw arcs intersecting each other at D.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.2 1
(iii) Join AD and BD.
(iv) Again with centre A and radius 4.5 cm and with centre B and radius 3 cm, draw arcs intersecting each other at C.
(v) Join AC and BC and also CD.
Then ABCD is the required quadrilateral.

Question 2.
Construct a quadrilateral ABCD in which BC = 7.5 cm, AC = AD = 6 cm, CD = 5 cm and BD = 10 cm.
Solution:
Steps of construction :
(i) Draw a line segment CD = 5 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.2 2
(ii) With centre C and D and radius 6 cm, draw line segments intersecting each other at A.
(iii) Join AC and AD.
(iv) Again with centre C and radius 7.5 cm and with centre D and radius 10 cm, draw arcs intersecting each other at B.
(v) Join CB, CA, DA, DB and AB.
Then ABCD is the required quadrilateral.

Question 3.
Construct a quadrilateral ABCD, when AB = 3 cm, CD = 3 cm, DA = 7.5 cm, AC = 8 cm and BD = 4 cm.
Solution:
Steps of construction :
This quadrilateral is not possible as
BD = 4 cm, AB = 3 cm and AD = 7.5 cm
The sum of any two sides of a triangle is greater than the third side.
But BD + AD = 4 + 3 = 7 cm
BD + AD < AD

Question 4.
Construct a quadrilateral ABCD given AD = 3.5 cm, BC = 2.5 cm, CD = 4.1 cm, AC = 7.3 cm and BD = 3.2 cm.
Solution:
Steps of construction :
(i) Draw a line segment CD = 4.1 cm.
(ii) With centre C and radius 7.3 cm and with centre D and radius 3.5 cm, draw arcs intersecting each other at A.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.2 3
(iii) Join AC and AD.
(iv) Again with centre C and radius 2.5 cm and with centre D and radius 3.2 cm, draw arcs intersecting each other at B.
(v) Join CB’, and DB’ and join AB’.
Then ABCD is the required quadrilateral.

Question 5.
Construct a quadrilateral ABCD given AD = 5 cm, AB = 5.5 cm, BC = 2.5 cm, AC = 7.1 cm and BD = 8 cm.
Solution:
Steps of construction:
(i) Draw a line segment AB = 5 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.2 4
(ii) With centre A and radius 7.1 cm and with centre B and radius 2.5 cm, draw arcs which intersect each other at C.
(iii) Join AC and BC.
(iv) Again with centre A and radius 5 cm and with centre B and radius 8 cm, draw arcs which intersect each other at D.
(v) Join AD and BD and CD.
Then ABCD is the required quadrilateral.

Question 6.
Construct a quadrilateral ABCD in which BC = 4 cm, CA = 5.6 cm, AD = 4.5 cm, CD = 5 cm and BD = 6.5 cm.
Solution:
Steps of construction:
(i) Draw a line segment CD = 5 cm.
RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.2 5
(ii) With centre C and radius 5.6 cm and with centre D and radius 4.5 cm, draw arcs which intersect each other at A.
(iii) Join AC and AD.
(iv) Again with centre C and radius 4 cm and with centre D and radius 6.5 cm, draw arcs which intersect each other at B.
(v) Join BC and BD and AB.
Then ABCD is the required quadrilateral.

Hope given RD Sharma Class 8 Solutions Chapter 18 Practical Geometry Ex 18.2 are helpful to complete your math homework.

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RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1

RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Linear Equations in One Variable Ex 9.1

Other Exercises

Solve each of the following equations and also verify your solution :
Question 1.
9\(\frac { 1 }{ 4 }\) = y – 1\(\frac { 1 }{ 3 }\)
Solution:
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 1
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 2
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 3

Question 2.
\(\frac { 5x }{ 3 }\) + \(\frac { 2 }{ 5 }\) = 1
Solution:
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 4

Question 3.
\(\frac { x }{ 2 }\) + \(\frac { x }{ 3 }\) + \(\frac { x }{ 4 }\) = 13
Solution:
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 5

Question 4.
\(\frac { x }{ 2 }\) + \(\frac { x }{ 8 }\) = \(\frac { 1 }{ 8 }\)
Solution:
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 6
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 7

Question 5.
\(\frac { 2x }{ 3 }\) – \(\frac { 3x }{ 8 }\) = \(\frac { 7 }{ 12 }\)
Solution:
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 8

Question 6.
(x + 2) (x + 3) + (x – 3) (x – 2) – 2x (x + 1) = 0
Solution:
(x + 2) (x + 3) + (x – 3) (x – 2) – 2x (x + 1) = 0
⇒ [x² + (2 + 3) x + 2 x 3] + [x² + (-3 – 2) x + (-3) (-2)] – 2x² – 2x = 0
⇒ x² + 5x + 6 + x² – 5x + 6 – 2x² – 2x = 0
⇒ x² + x² – 2x² + 5x – 5x – 2x + 6 + 6 = 0
⇒ -2x + 12 = 0
Subtracting 12 from both sides,
-2x + 12 – 12 = 0 – 12
⇒ -2x = -12
Dividing by -2,
x = 6
Verification:
L.H.S. = (x + 2) (x + 3) + (x – 3) (x – 2) – 2x (x + 1)
= (6 + 2) (6 + 3) + (6 – 3) (6 – 2) – 2 x 6 (6 + 1)
= 8 x 9 + 3 x 4 – 12 x 7
= 72 + 12 – 84
= 84 – 84
= 0
= R.H.S.

Question 7.
\(\frac { x }{ 2 }\) – \(\frac { 4 }{ 5 }\) + \(\frac { x }{ 5 }\) +\(\frac { 3x }{ 10 }\) = \(\frac { 1 }{ 5 }\)
Solution:
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 9
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 10

Question 8.
\(\frac { 7 }{ x }\) + 35 = \(\frac { 1 }{ 10 }\)
Solution:
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 11
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 12

Question 9.
\(\frac { 2x – 1 }{ 3 }\) – \(\frac { 6x – 2 }{ 5 }\) = \(\frac { 1 }{ 3 }\)
Solution:
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 13
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 14
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 15

Question 10.
13 (y – 4) – 3 (y – 9) – 5 (y + 4) = 0
Solution:
13 (y – 4) – 3 (y – 9) – 5 (y + 4) = 0
⇒ 13y – 52 – 3y + 27 – 5y – 20 = 0
⇒ 13y – 3y – 5y – 52 + 27 – 20 = 0
⇒ 13y – 8y – 72 + 27 = 0
⇒ 5y – 45 = 0
Dividing by 5,
y = 9
Verification:
L.H.S. = 13 (y – 4) – 3 (y – 9) – 5 (y + 4)
= 13 (9 – 4) – 3 (9 – 9) – 5 (9 + 4)
= 13 x 5 – 3 x 0 – 5 x 13
= 65 – 0 – 65
= 0
= R.H.S.

Question 11.
\(\frac { 2 }{ 3 }\) (x – 5) – \(\frac { 1 }{ 4 }\) (x – 2) = \(\frac { 9 }{ 2 }\)
Solution:
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 16
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 17
RD Sharma Class 8 Solutions Chapter 9 Linear Equations in One Variable Ex 9.1 18

Hope given RD Sharma Class 8 Solutions Linear Equations in One Variable Ex 9.1 are helpful to complete your math homework.

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RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4

RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4

Other Exercises

Find the following products 

Question 1.
2a3 (3a + 5b)
Solution:
2a3 (3a + 5b) = 2a3 x 3a + 2a3 x 5b
= 6a3 +1 + 10a3b
= 6a4 + 10a3b

Question 2.
-11a (3a + 2b)
Solution:
-11a (3a + 2b) = -11a x 3a – 11a x 2b
= -33a2– 22ab

Question 3.
-5a (7a – 2b)
Solution:
-5a (7a – 2b) = -5a x 7a- 5a x (-2b)
= -35a2 + 10ab

Question 4.
-11y2 (3y + 7)
Solution:
-11y2 (3y + 7) = -11y2 x 3y – 11y2 x 7
= -33y2+1-77y2
= 33y3-77y2

Question 5.
\(\frac { 6x }{ 5 }\) (x3+y3)
Solution:
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 1

Question 6.
xy (x3-y3)
Solution:
xy (x3 – y3) =xy x x3 – xy x y3
= x1 + 3 x y – x x y1+3
= x4y – xy4

Question 7.
0.1y (0.1x5 + 0.1y)
Solution:
0.1y (0.1x5 + 0.1y) = 0.1y x 0.1x5 + 0.1y x 0.1y
= 0.01x5y + 0.01y2

Question 8.
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 2
Solution:
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 3

Question 9.
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 4
Solution:
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 5

Question 10.
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 6
Solution:
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 7

Question 11.
5x (10x2y – 100xy2)
Solution:
5x (10x2y – 100xy2)
= 1.5x x 10x2y – 1.5x x 100xy2
= 15x1 + 2y- 150x1+1 x y2
15 x3y- 150xy2

Question 12.
4.1xy (1.1x-y)
Solution:
4.1xy (1.1x-y) = 4.1xy x 1.1x – 4.1xy x y
= 4.51x2y-4.1xy2

Question 13.
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 8
Solution:
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 9

Question 14.
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 10
Solution:
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 11

Question 15.
\(\frac { 4 }{ 3 }\) a (a2 + 62 – 3c2)
Solution:
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 12
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 13

Question 16.
Find the product 24x2 (1 – 2x) and evaluate its value for x = 3.
Solution:
24x2 (1 – 2x) = 24x2 x 1 + 24x2 x (-2x)
= 24x2 + (-48x2+1)
= 24x2 – 48x3
If x = 3, then
= 24 (3)2 – 48 (3)3
= 24 x 9-48 x 27 = 216- 1296
= -1080

Question 17.
Find the product of -3y (xy +y2) and find its value for x = 4, and y = 5.
Solution:
-3y (xy + y2) = -3y x xy – 3y x y2
= -3xy2 -3y2 +1  = -3xy2 – 3y3
If x = 4, y = 5, then
= -3 x 4 (5)2 – 3 (5)3 = -12 x 25 – 3 x 125
= -300 – 375 = – 675

Question 18.
Multiply – \(\frac { 3 }{ 2 }\) x2y3 by (2x-y) and verify the answer for x = 1 and y = 2.
Solution:
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 14
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 15

Question 19.
Multiply the monomial by the binomial and find the value of each for x = -1, y = 25 and z =05 :
(i) 15y2 (2 – 3x)
(ii) -3x (y2 + z2)
(iii) z2 (x – y)
(iv) xz (x2 + y2)
Solution:
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 16
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 17

Question 20.
Simplify :
(i) 2x2 (at1 – x) – 3x (x4 + 2x) -2 (x4 – 3x2)
(ii) x3y (x2 – 2x) + 2xy (x3 – x4)
(iii) 3a2 + 2 (a + 2) – 3a (2a + 1)
(iv) x (x + 4) + 3x (2x2 – 1) + 4x2 + 4
(v) a (b-c) – b (c – a) – c (a – b)
(vi) a (b – c) + b (c – a) + c (a – b)
(vii) 4ab (a – b) – 6a2 (b – b2) -3b2 (2a2 – a) + 2ab (b-a)
(viii) x2 (x2 + 1) – x3 (x + 1) – x (x3 – x)
(ix) 2a2 + 3a (1 – 2a3) + a (a + 1)
(x) a2 (2a – 1) + 3a + a3 – 8
(xi) \(\frac { 3 }{ 2 }\)-x2 (x2 – 1) + \(\frac { 1 }{4 }\)-x2 (x2 + x) – \(\frac { 3 }{ 4 }\)x (x3 – 1)
(xii) a2b (a – b2) + ab2 (4ab – 2a2) – a3b (1 – 2b)
(xiii )a2b (a3 – a + 1) – ab (a4 – 2a2 + 2a) – b (a3– a2 -1)
Solution:
(i) 2x2 (x3 -x) – 3x (x4 + 2x) -2 (x4 – 3x2)
= 2xx x3-2x2x x-3x x x4-3x x 2x-2x4 + 6x2
= 2x2 + 3– 2x2 +1 – 3x,1+ 4-6x,1+1 -2x4 + 6x2
= 2x5 – 2x3 – 3x5 — 6x2 – 2x4 + 6x2
= 2x5 – 3x5 – 2a4 – 2x3 + 6x2 – 6x2
= -x5 – 2x4 – 2x3 + 0
= -x5-2x4-2x3
(ii) x3y (x2 – 2x) + 2xy (x3 – x4)
= x3y x x2 – x3y x 2x + 2ay x ac3 – 2xy x x4
= x3 + 2y-2x3 + 1 y + 2x1 + 3y – 2yx4+1
= x5y – 2x4y + 2x4y – 2yx5
= -x5y
(iii) 3a2 + 2 (a + 2) – 3a (2a + 1)
= 3a2 + 2a + 4 – 6a2 – 3a
= 3a2 – 6a2 + 2a – 3a + 4
= -3a2 – a + 4
(iv) x (x + 4) + 3x (2x2 – 1) + 4x2 + 4
= x2 + 4x + 3x x 2x2 – 3x x 1 + 4x2 + 4
= x2 + 4x + 6x2 +1 – 3x + 4x2 + 4
= x2 + 4x + 6x3 – 3x + 4x2 + 4
= 6a3 + 4x2 + x2 + 4x – 3x + 4
= 6x3 + 5x2 + x + 4
(v) a (b – c)-b (c – a) – c (a – b)
= ab – ac – be + ab – ac + bc
= 2ab – 2ac
(vi) a (b – c) + b (c – a) + c (a – b)
= ab – ac + bc – ab + ac – bc
= ab – ab + bc – be + ac – ac
= 0
(vii) 4ab (a – b) – 6a2 (b – b2) -3b2 (2a2 – a) + 2ab (b – a)
= 4a2b – 4ab2 – 6a2b + 6a2b2 – 6a2b2 + 3ab2 + 2ab2 – 2a2b
= 4a2b- 6a2b – 2 a2b – 4ab2 + 3 ab2 + 2ab2 + 6a2b2 – 6a2b2
= 4a2b – 8a2b – 4ab2 + 5 ab2 + 0
= – 4a2b + ab2
(viii) x2 (x2 + 1) – x3 (x + 1) – x (x3 – x)
= x2 + 2 + x2 – x3 + 1 – x3 – x1 + 3 + x1 + 1
= x4 + x2-x4-x3-x4 + x2
= x4-x4-x4-x3 + x2 + x2
= -x4 – x3 + 2x2
(ix) 2a2 + 3a (1 – 2a3) + a (a + 1)
= 2a2 + 3 a – 3 a x 2a3 + a2 + a
= 2a2 + 3a – 6a1 + 3 + a2 + a
= 2a2 + 3a – 6a4 + a2 + a
= -6a4 + 3a2 + 4a
(x) a2 (2a – 1) + 3a + a3 – 8
= 2 a2 x a – a2 x 1+3a + a3-8
= 2a3 – a2 + 3a + a3 – 8
= 2a3 + a3 – a2 + 3a – 8
= 3a3 – a2 + 3a – 8
RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 18
(xii) a2b (a – b2) + ab2 (4ab – 2a2) – a3b (1 – 2b)
= a2b x a – a2b x b1 + ab2 x 4ab – ab1 x2a2 -a3b x 1 + a3b x 2b
= a2+1 b-a2b2 +1+ 4a1 +1 b2 +1 -2a2+1 b2-a3b + 2a3b1 +1
= a’b – a2b3 + 4a2b3 – 2a3b2 – a3b + 2a3b2
= a3b – a3b – a2b3 + 4a2b3 – 2a3b2 + 2a3b2
= 0 + 3a2b3 + 0 = 3 a2b3
(xiii) a2b (a3 – a + 1) – ab (a4 – 2a2 + 2a) – b (a3 -a2– 1)
= a2b x a3 – a2b x a + a2b – ab x a2 + ab x 2a2 – ab x 2a- ba3 + ba2 + b
= a2+ 3b – a2+1 b + a2b -a1 + 4b + 2a1 + 2b- 2a1+1 b- a3b + a2b + b
= a5b – a3b + a26 – a5b + 2a3b – 2a2b – a3b + a2b + b
= a5b – a3b + 2a3b – a36 – a3b + a2b – 2a2b + a2b + b
= a3b – a5b + 2a3b – 2a3b + 2a2b-2a2b + b
= 0 + 0 + 0 + b = b

Hope given RD Sharma Class 8 Solutions Chapter 6 Algebraic Expressions and Identities Ex 6.4 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1

RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II (Quadrilaterals) Ex 16.1

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1

Question 1.
Define the following terms:
(i) Quadrilateral
(ii) Convex Quadrilateral.
Solution:
(i) Quadrilateral: A closed figure made of four line segments is called a quadrilateral such that:
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 1
(a) no three points of them are collinear
(b) the line segments do not intersect except at their ends points.
(ii) Convex quadrilateral: A quadrilateral is called a convex quadrilateral of the line containing any side of the quadrilateral has the remaining vertices on the same side of it. In the figure, quadrilateral ABCD is a convex quadrilateral.
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 2

Question 2.
In a quadrilateral, define each of the following:
(i) Sides
(ii) Vertices
(iii) Angles
(iv) Diagonals
(v) Adjacent angles
(vi) Adjacent sides
(vii) Opposite sides
(viii) Opposite angles
(ix) Interior
(x) Exterior
Solution:
(i) Sides: In a quadrilateral ABCD, form line segments AB, BC, CD and DA are called sides of the quadrilateral.
(ii) Vertices : The ends points are called the vertices of the quadrilateral. Here in the figure, A, B, C and D are its vertices.
(iii) Angles: A quadrilateral has four angles which are at their vertices. In the figure, ∠A, ∠B, ∠C and ∠D are its angles.
(iv) Diagonals: The line segment joining the opposite vertices is called diagonal. A quadrilateral has two diagonals.
(v) Adjacent Angles : The angles having a common arm (side) are called adjacent angles.
(vi) Adjacent sides : If two sides of a quadrilateral have a common end-point, these are called adjacent sides.
(vii) Opposite sides: If two sides do not have a common end-point of a quadrilateral, they are called opposite sides.
(viii) Opposite angles : The angles which are not adjacent are called opposite angles.
(ix) Interior: The region which is surrounded by the sides of the quadrilateral is called its interior.
(x) Exterior : The part of the plane made up by all points as the not enclosed by the quadrilateral, is called its exterior.
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 3

Question 3.
Complete each of the following, so as to make a true statement:
(i) A quadrilateral has ………… sides.
(ii) A quadrilateral has ………… angles.
(iii) A quadrilateral has ……….. vertices, no three of which are …………
(iv) A quadrilateral has …………. diagonals.
(v) The number of pairs of adjacent angles of a quadrilateral is ………….
(vi) The number of pairs of opposite angles of a quadrilateral is ……………
(vii) The sum of the angles of a quadrilateral is …………
(viii) A diagonal of a quadrilateral is a line segment that joins two ………. vertices of the quadrilateral.
(ix) The sum of the angles of a quadrilateral is …………. right angles.
(x) The measure of each angle of a convex quadrilateral is …………. 180°.
(xi) In a quadrilateral the point of intersection of the diagonals lies in ………….. of the quadrilateral.
(xii) A point is in the interior of a convex quadrilateral, if it is in the ……….. of its two opposite angles.
(xiii) A quadrilateral is convex if for each side, the remaining …………. lie on the same side of the line containing the side.
Solution:
(i) A quadrilateral has four sides.
(a) A quadrilateral has four angles.
(iii) A quadrilateral has four vertices, no three of which are collinear .
(iv) A quadrilateral has two diagonals.
(v) The number of pairs of adjacent angles of a quadrilateral is four .
(vi) The number of pairs of opposite angles ot a quadrilateral is two.
(vii) The sum of the angles of a quadrilateral is 360°.
(viii) A diagonal of a quadrilateral is a line segment that join two opposite vertices of the quadrilateral.
(ix) The sum of the angles of a quadrilateral is 4 right angles.
(x) The measure of each angle of a convex quadrilateral is less than 180°.
(xi) In a quadrilateral the point of intersection of the diagonals lies in interior of the quadrilateral.
(xii) A point is in the interior of a convex quadrilateral, if it is in the interior of its two opposite angles.
(xiii) A quadrilateral is convex if for each side, the remaining vertices lie on the same side of the line containing the side.

Question 4.
In the figure, ABCD is a quadrilateral.
(i) Name a pair of adjacent sides.
(ii) Name a pair of opposite sides.
(iii) How many pairs of adjacent sides are there?
(iv) How many pairs of Opposite sides are there ?
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 4
(v) Name a pair of adjacent angles.
(vi) Name a pair of opposite angles.
(vii) How many pairs of adjacent angles are there ?
(viii) How many pairs of opposite angles are there ?
Solution:
In the figure, ABCD is a quadrilateral
(i) Pairs of adjacent sides are AB, BC, BC, CD, CD, DA, DA, AB.
(ii) Pairs of opposite sides are AB and CD; BC and AD.
(iii) There are four pairs of adjacent sides.
(iv) There are two pairs of opposite sides.
(v) Pairs of adjacent angles are ∠A, ∠B; ∠B, ∠C; ∠C, ∠D; ∠D, ∠A.
(vi) Pairs of opposite angles are ∠A and ∠C; ∠B and ∠D.
(vii) There are four pairs of adjacent angles.
(viii) There are two pairs of opposite angles.

Question 5.
The angles of a quadrilateral are 110°, 72°, 55° and x°. Find the value of x.
Solution:
Sum of four angles of quadrilateral is 360°
110° + 12° + 55° + x° = 360°
⇒ 237° + x° = 360°
⇒ x° = 360° – 237° = 123°
x = 123°

Question 6.
The three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angle.
Solution:
The sum of four angles of a quadrilateral = 360°
Three angles are 110°, 50° and 40°
Let fourth angle = x
Then 110° + 50° + 40° + x° = 360°
⇒ 200° + x° = 360°
⇒ x = 360° – 200° = 160°
x = 160°

Question 7.
A quadrilateral has three acute angles each measures 80°. What is the measure of fourth angle ?
Solution:
Sum of four angles of a quadrilateral = 360°
Sum of three angles having each angle equal to 80° = 80° x 3 = 240°
Let fourth angle = x
Then 240° + x = 360°
⇒ x° = 360° – 240°
⇒ x° = 120°
Fourth angle = 120°

Question 8.
A quadrilateral has all its four angles of the same measure. What is the measure of each ?
Solution:
Let each equal angle of a quadrilateral = x
4x° = 360°
⇒ x° = \(\frac { 360 }{ 4 }\) = 90°
Each angle will be = 90°

Question 9.
Two angles of a quadrilateral are of measure 65° and the other two angles are equal. What is the measure of each of these two angles ?
Solution:
Measures of two angles each = 65°
Sum of these two angles = 2 x 65°= 130°
But sum of four angles of a quadrilateral = 360°
Sum of the remaining two angles = 360° – 130° = 230°
But these are equal to each other
Measure of each angle = \(\frac { 230 }{ 2 }\) = 115°

Question 10.
Three angles of a quadrilateral are equal. Fourth angle is of measure 150°. What is the measure of equal angles ?
Solution:
Sum of four angles of a quadrilateral = 360°
One angle = 150°
Sum of remaining three angles = 360° – 150° = 210°
But these three angles are equal
Measure of each angle = \(\frac { 210 }{ 3 }\) = 70°

Question 11.
The four angles of a quadrilateral are as 3 : 5 : 7 : 9. Find the angles.
Solution:
Sum of four angles of a quadrilateral = 360°
and ratio in angles = 3 : 5 : 7 : 9
Let first angles = 2x
Then second angle = 5x
third angle = 7x
and fourth angle = 9x
3x + 5x + 7x + 9x = 360°
⇒ 24x = 369°
⇒ x = \(\frac { 360 }{ 24 }\) = 15°
First angle = 3x = 3 x 15° = 45°
second angle = 5x = 5 x 15° = 75°
third angle = 7x = 7 x 15° = 105°
and fourth angle = 9x = 9 x 15° = 135°

Question 12.
If the sum of the two angles of a quadrilateral is 180°, what is the sum of the remaining two angles ?
Solution:
Sum of four angles of a quadrilateral = 360°
and sum of two angle out of these = 180°
Sum of other two angles will be = 360° – 180° = 180°

Question 13.
In the figure, find the measure of ∠MPN.
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 5
Solution:
In the figure, OMPN is a quadrilateral in which
∠O = 45°, ∠M = ∠N = 90° (PM ⊥ OA and PN ⊥ OB)
Let ∠MPN = x°
∠O + ∠M + ∠N + ∠MPN = 360° (Sum of angles of a quadrilateral)
⇒ 45° + 90° + 90° + x° = 360°
⇒ 225° + x° = 360°
⇒ x° = 360° – 225°
⇒x = 135°
∠MPN = 135°

Question 14.
The sides of a quadrilateral are produced in order. What is the sum of the four exterior angles ?
Solution:
The sides of a quadrilateral ABCD are produced in order, forming exterior angles ∠1, ∠2, ∠3 and ∠4.
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 6
Now ∠DAB + ∠1 = 180° (Linear pair) ……(i)
Similarly,
∠ABC + ∠2 = 180°
∠BCD + ∠3 = 180°
and ∠CDA + ∠4 = 180°
Adding, we get
∠DAB + ∠1 + ∠ABC + ∠2 + ∠BCD + ∠3 + ∠CDA + ∠4 = 180° + 180° + 180° + 180° = 720°
⇒ ∠DAB + ∠ABC + ∠CDA + ∠ADC + ∠1 + ∠2 + ∠3 + ∠4 = 720°
But ∠DAB + ∠ABC + ∠CDA + ∠ADB = 360° (Sum of angles of a quadrilateral)
360° + ∠1 + ∠2 + ∠3 + ∠4 = 720°
⇒ ∠l + ∠2 + ∠3 + ∠4 = 720° – 360° = 360°
Sum of exterior angles = 360°

Question 15.
In the figure, the bisectors of ∠A and ∠B meet at a point P. If ∠C = 100° and ∠D = 50°, find the measure of ∠APB.
Solution:
In quadrilateral ABCD,
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 7
∠D = 50°, ∠C = 100°
PA and PB are the bisectors of ∠A and ∠B.
In quadrilateral ABCD,
∠A + ∠B + ∠C + ∠D = 360° (Sum of angles of a quadrilateral)
⇒ ∠A + ∠B + 100° + 50° = 360°
⇒ ∠A + ∠B + 150° = 360°’
⇒ ∠A + ∠B = 360° – 150° = 210°
and \(\frac { 1 }{ 2 }\) ∠A + \(\frac { 1 }{ 2 }\) ∠B = \(\frac { 210 }{ 2 }\) = 105°
(PA and PB are bisector of ∠A and ∠B respectively)
∠PAB + ∠PBA = 105°
⇒ ∠PAB + ∠PBA + ∠APB = 180° (Sum of angles of a triangle)
⇒ 105° + ∠APB = 180°
⇒ ∠APB = 180° – 105° = 75°
∠APB = 75°

Question 16.
In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1 : 2 : 4 : 5. Find the measure of each angle of the quadrilateral.
Solution:
Sum of angles A, B, C and D of a quadrilateral = 360°
i.e. ∠A + ∠B + ∠C + ∠D = 360°
But ∠A = ∠B = ∠C = ∠D = 1 : 2 : 4 : 5
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 8
Let ∠A = x,
Then ∠B = 2x
∠C = 4x
∠D = 5x
x + 2x + 4x + 5x = 360°
⇒ 12x = 360°
⇒ x = \(\frac { 360 }{ 12 }\) = 30°
∠A = x = 30°
∠B = 2x = 2 x 30° = 60°
∠C = 4x = 4 x 30° = 120°
∠D = 5A = 5 x 30° = 150°

Question 17.
In a quadrilateral ABCD, CO and DO are the bisectors of ∠C and ∠D respectively. Prove that ∠COD = \(\frac { 1 }{ 2 }\) (∠A + ∠B).
Solution:
In quadrilateral ABCD,
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 9
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 10

Question 18.
Find the number of sides of a regular polygon when each of its angles has a measures of
(i) 160°
(ii) 135°
(iii) 175°
(iv) 162°
(v) 150°.
Solution:
In a n-sided regular polygon, each angle
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 11
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 12
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 13
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 14

Question 19.
Find the number of degrees in each exterior angle of a regular pentagon.
Solution:
In a pentagon or a polygon, sum of exterior angles formed by producing the sides in order, is four right angles or 360°
Each exterior angle = \(\frac { 360 }{ 5 }\) = 72°

Question 20.
The measure of angles of a hexagon are x°, (x – 5)° (x – 5)°, (2x – 5)°, (2x – 5)°, (2x + 20)°. Find the value of x.
Solution:
We know that the sum of interior angels of a hexagon = 720° (180° x 4)
⇒ x + x – 5 + x – 5 + 2x – 5 + 2x – 5 + 2x + 20 = 720°
⇒ 9x – 20 + 20 = 720
⇒ 9x = 720
⇒ x = \(\frac { 720 }{ 9 }\) = 80°
x = 80°

Question 21.
In a convex hexagon, prove that the sum of all interior angles is equal to twice the sum of its exterior angles formed by producing the sides in the same order.
Solution:
In a convex hexagon ABCDEF, its sides AB, BG, CD, DE, EF and FA are produced in order forming exterior angles a, b, c, d, e, f
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 15
∠a + ∠b + ∠c + ∠d + ∠e + ∠f = 4 right angles (By definition)
By joining AC, AD, and AE, 4 triangles ABC, ACD, ADE and AEF are formed
In ∆ABC,
∠1 + ∠2 + ∠3 = 180° = 2 right angle (Sum of angles of a triangle) …… (i)
Similarly,
In ∆ACD,
∠4 +∠5 + ∠6 = 180° = 2 right angles
In ∆ADE,
∠1 + ∠8 + ∠9 = 2 right angles …(iii)
In ∆AEF,
∠10 + ∠11 + ∠12 = 2 right angles …(iv)
Joining (i), (ii), (iii) and (iv)
∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8 + ∠9 + ∠10 + ∠11 + ∠12 = 8 right angles
⇒ ∠2 + ∠3 + ∠5 + ∠6 + ∠8 + ∠9 + ∠11 + ∠12 + ∠1 + ∠4 + ∠7 + ∠10 = 8 right angles
⇒ ∠B + ∠C + ∠D + ∠E +∠F + ∠A = 8 right angles
⇒ ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = 2 (∠a + ∠b + ∠c + ∠d + ∠e + ∠f)
Sum of all interior angles = 2(the sum of exterior angles)
Hence proved.

Question 22.
The sum of the interior angles of a polygon is three times the sum of its exterior angles. Determine the number of sides of the polygon.
Solution:
Let number of sides of a regular polygon = n
Each interior angle = \(\frac { 2n – 4 }{ n }\) right angles
Sum of all interior angles = \(\frac { 2n – 4 }{ n }\) x n
right angles = (2n – 4) right angles
But sum of exterior angles = 4 right angles
According to the condition,
(2n – 4) = 3 x 4 (in right angles)
⇒ 2n – 4 = 12
⇒ 2n = 12 + 4 = 16
⇒ n = 8
Number of sides of the polygon = 8

Question 23.
Determine the number of sides of a polygon whose exterior and interior angles are in the ratio 1 : 5.
Solution:
Ratio in exterior angle and interior angles of a regular polygon = 1 : 5
But sum of interior and exterior angles = 180° (Linear pair)
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 16
By cross multiplication:
6n – 12 = 5n
⇒ 6n – 5n = 12
⇒ n = 12
Number of sides of polygon is 12

Question 24.
PQRSTU is a regular hexagon. Determine each angle of ∆PQT.
Solution:
In regular hexagon, PQRSTU, diagonals PT and QT are joined.
RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 17
In ∆PUT, PU = UT
∠UPT = ∠UTP
But ∠UPT + ∠UTP = 180° – ∠U = 180° – 120° = 60°
∠UPT = ∠UTP = 30°
∠TPQ = 120° – 30° = 90° (QT is diagonal which bisect ∠Q and ∠T)
∠PQT = \(\frac { 120 }{ 2 }\) = 60°
Now in ∆PQT,
∠TPQ + ∠PQT + ∠PTQ = 180° (Sum of angles of a triangle)
⇒ 90° + 60° + ∠PTQ = 180°
⇒ 150° + ∠PTQ = 180°
⇒ ∠PTQ = 180° – 150° = 30°
Hence in ∆PQT,
∠P = 90°, ∠Q = 60° and ∠T = 30°

Hope given RD Sharma Class 8 Solutions Chapter 16 Understanding Shapes II Ex 16.1 are helpful to complete your math homework.

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RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1

RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I (Polygons) Ex 15.1

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1

Question 1.
Draw rough diagrams to illustrate the following:
(i) Open curve
(ii) Closed curve
Solution:
RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 1

Question 2.
Classify the following curves as open or closed.
RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 2
Solution:
Open curves : (i), (iv) and (v) are open curves.
(ii) , (iii), and (vi) are closed curves.

Question 3.
Draw a polygon and shade its interior. Also draw its diagonals, if any.
Solution:
In the given polygon, the shaded portion is its interior region AC and BD are the diagonals of polygon ABCD.
RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 3

Question 4.
Illustrate, if possible, each one of the following with a rough diagram:
(i) A closed curve that is not a polygon.
(ii) An open curve made up entirely of line segments.
(iii) A polygon with two sides.
Solution:
(i) Close curve but not a polygon.
RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 4
(ii) An open curve made up entirely of line segments.
RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 5
(iii) A polygon with two sides. It is not possible. At least three sides are necessary

Question 5.
Following are some figures : Classify each of these figures on the basis of the following:
RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 6
RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 7
(i) Simple curve
(ii) Simple closed curve
(iii) Polygon
(iv) Convex polygon
(v) Concave polygon
(vi) Not a curve
Solution:
(i) It is a simple closed curve and a concave polygon.
(ii) It is a simple closed curve and convex polygon.
(iii) It is neither a curve nor polygon.
(iv) it is neither a curve not a polygon.
(v) It is a simple closed curve but not a polygon.
(vi) It is a simple closed curve but not a polygon.
(vii) It is a simple closed curve but not a polygon.
(viii) It is a simple closed curve but not a polygon.

Question 6.
How many diagonals does each of the following have ?
(i) A convex quadrilateral
(ii) A regular hexagon
(iii) A triangle.
Solution:
(i) A convex quadrilateral
Here n = 4
RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 8
RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 9

Question 7.
What is a regular polygon ? State the name of a regular polygon of:
(i) 3 sides
(ii) 4 sides
(iii) 6 sides.
Solution:
A regular polygon is a polygon which has all its sides equal and so all angles are equal,
(i) 3 sides : It is an equilateral triangle.
(ii) 4 sides : It is a square.
(iii) 6 sides : It is a hexagon.

Hope given RD Sharma Class 8 Solutions Chapter 15 Understanding Shapes I Ex 15.1 are helpful to complete your math homework.

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RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5

RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5

Other Exercises

Question 1.
Mr. Cherian purchased a boat for Rs 16,000. If the total cost of the boat is depreciating at the rate of 5% per annum, calculate its value after 2 years.
Solution:
Cost of boat = Rs 16,000
Rate of depreciating = 5% p.a.
Period = 2 years
Value of boat after 2 years
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 1

Question 2.
The value of a machine depreciates at the rate of 10% per annum. What will be its value 2 years hence, if the present value is Rs 1,0,000 ? Also, find the total depreciation during this period.
Solution:
Present value of machine = Rs 1,00,000
Rate of depreciation = 10% p.a.
Period (n) = 2 years
Value of machine after 2 years
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 2

Question 3.
Pritam bought a plot of land for Rs 6,40,000. Its value is increasing by 5% of its previous value after every six months. What will be the value of the plot after 2 years ?
Solution:
Present value of plot = Rs 6,40,000
Increase = 5% per half year
Period (n) = 2 years or 4 half years
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 3
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 4

Question 4.
Mohan purchased a house for Rs 30,000 and its value is depreciating at the rate of 25% per year. Find the value of the house after 3 years.
Solution:
Present value of the house (P) = Rs 30,000
Rate of depreciation = 25% p.a.
Period (n) = 3 years
Value of house after 3 years
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 5

Question 5.
The value of a machine depreciates at the rate of 10% per annum. It was purchased 3 years ago. If its present value is Rs 43,740, find its purchased price.
Solution:
Let the purchase price of machine = Rs P
Rate of depreciation = 10% p.a.
Period (n) = 3 years.
and present value = Rs 43,740
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 6
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 7

Question 6.
The value of a refrigerator which was purchased 2 years ago, depreciates at 12% per annum. If its present value is Rs 9,680, for how much was it purchased ?
Solution:
Let the refrigerator was purchased for = Rs P
Rate of depreciation (R) = 12% p.a.
Period (n) = 2 years
and present value (A) = Rs 9,680
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 8

Question 7.
The cost of a TV set was quoted Rs 17,000 at the beginning of 1999. In the beginning of2000, the price was hiked by 5%. Because of decrease in demand the cost was reduced by 4% in the beginning of 2001. What is the cost of the TV set in 2001 ?
Solution:
List price of TV set in 1999 = Rs 17,000
Rate of hike in 2000 = 5%
Rate of decrease in 2001 = 4%
Price of TV set in 2001
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 9

Question 8.
Ashish started the business with an initial investment of Rs 5,00,000. In the first year, he incurred a loss of 4%. However, during the second year he earned a profit of 5% which in third year, rose to 10%. Calculate the net profit for the entire period of 3 years.
Solution:
Initial investment = Rs 5,00,000
In the first year, rate of loss = 4%
In the second year, rate of gain = 5%
and in the third year, rate of gain = 10%
Investment after 3 years
RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 10

Hope given RD Sharma Class 8 Solutions Chapter 14 Compound Interest Ex 14.5 are helpful to complete your math homework.

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