Understanding Shapes – III (Special Types of Quadrilaterals) - EXERCISE 17.2 | Mathematics (Maths) Class 8 PDF Download

 Page 1


 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
EXERCISE 17.3                                                  PAGES NO: 17.22 
 
1. Which of the following statements are true for a rectangle? 
(i) It has two pairs of equal sides. 
(ii) It has all its sides of equal length. 
(iii) Its diagonals are equal. 
(iv) Its diagonals bisect each other. 
(v) Its diagonals are perpendicular. 
(vi) Its diagonals are perpendicular and bisect each other. 
(vii) Its diagonals are equal and bisect each other. 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
(ix) All rectangles are squares. 
(x) All rhombuses are parallelograms. 
(xi) All squares are rhombuses and also rectangles. 
(xii) All squares are not parallelograms. 
Solution: 
(i) It has two pairs of equal sides. 
 True, in a rectangle two pairs of sides are equal. 
 
(ii) It has all its sides of equal length. 
False, in a rectangle only two pairs of sides are equal. 
 
(iii) Its diagonals are equal. 
True, in a rectangle diagonals are of equal length. 
 
(iv) Its diagonals bisect each other. 
True, in a rectangle diagonals bisect each other. 
 
(v) Its diagonals are perpendicular. 
False, Diagonals of a rectangle need not be perpendicular. 
 
(vi) Its diagonals are perpendicular and bisect each other. 
False, Diagonals of a rectangle need not be perpendicular. Diagonals only bisect each 
other. 
 
(vii) Its diagonals are equal and bisect each other. 
True, Diagonals are of equal length and bisect each other. 
 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
Page 2


 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
EXERCISE 17.3                                                  PAGES NO: 17.22 
 
1. Which of the following statements are true for a rectangle? 
(i) It has two pairs of equal sides. 
(ii) It has all its sides of equal length. 
(iii) Its diagonals are equal. 
(iv) Its diagonals bisect each other. 
(v) Its diagonals are perpendicular. 
(vi) Its diagonals are perpendicular and bisect each other. 
(vii) Its diagonals are equal and bisect each other. 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
(ix) All rectangles are squares. 
(x) All rhombuses are parallelograms. 
(xi) All squares are rhombuses and also rectangles. 
(xii) All squares are not parallelograms. 
Solution: 
(i) It has two pairs of equal sides. 
 True, in a rectangle two pairs of sides are equal. 
 
(ii) It has all its sides of equal length. 
False, in a rectangle only two pairs of sides are equal. 
 
(iii) Its diagonals are equal. 
True, in a rectangle diagonals are of equal length. 
 
(iv) Its diagonals bisect each other. 
True, in a rectangle diagonals bisect each other. 
 
(v) Its diagonals are perpendicular. 
False, Diagonals of a rectangle need not be perpendicular. 
 
(vi) Its diagonals are perpendicular and bisect each other. 
False, Diagonals of a rectangle need not be perpendicular. Diagonals only bisect each 
other. 
 
(vii) Its diagonals are equal and bisect each other. 
True, Diagonals are of equal length and bisect each other. 
 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
False, Diagonals are of equal length and bisect each other. Diagonals of a rectangle need 
not be perpendicular 
 
(ix) All rectangles are squares. 
False, in a square all sides are of equal length. 
 
(x) All rhombuses are parallelograms. 
True, all rhombuses are parallelograms, since opposite sides are equal and parallel. 
 
(xi) All squares are rhombuses and also rectangles. 
True, all squares are rhombuses, since all sides are equal in a square and rhombus. All 
squares are rectangles, since opposite sides are equal and parallel. 
 
(xii) All squares are not parallelograms. 
False, all squares are parallelograms, since opposite sides are parallel and equal. 
 
2. Which of the following statements are true for a square? 
(i) It is a rectangle. 
(ii) It has all its sides of equal length. 
(iii) Its diagonals bisect each other at right angle. 
(v) Its diagonals are equal to its sides. 
Solution: 
(i) It is a rectangle. 
True. Since, opposite sides are equal and parallel where, each angle is right angle. 
 
(ii) It has all its sides of equal length. 
True. Since, ides of a square are of equal length. 
 
(iii) Its diagonals bisect each other at right angle. 
True. Since, diagonals of a square bisect each other at right angle. 
 
(v) Its diagonals are equal to its sides. 
False. Since, diagonals of a square are of equal length. Length of diagonals is not equal to 
the length of sides 
 
3. Fill in the blanks in each of the following, so as to make the statement true : 
(i) A rectangle is a parallelogram in which ________. 
(ii) A square is a rhombus in which __________. 
(iii) A square is a rectangle in which ___________. 
Page 3


 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
EXERCISE 17.3                                                  PAGES NO: 17.22 
 
1. Which of the following statements are true for a rectangle? 
(i) It has two pairs of equal sides. 
(ii) It has all its sides of equal length. 
(iii) Its diagonals are equal. 
(iv) Its diagonals bisect each other. 
(v) Its diagonals are perpendicular. 
(vi) Its diagonals are perpendicular and bisect each other. 
(vii) Its diagonals are equal and bisect each other. 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
(ix) All rectangles are squares. 
(x) All rhombuses are parallelograms. 
(xi) All squares are rhombuses and also rectangles. 
(xii) All squares are not parallelograms. 
Solution: 
(i) It has two pairs of equal sides. 
 True, in a rectangle two pairs of sides are equal. 
 
(ii) It has all its sides of equal length. 
False, in a rectangle only two pairs of sides are equal. 
 
(iii) Its diagonals are equal. 
True, in a rectangle diagonals are of equal length. 
 
(iv) Its diagonals bisect each other. 
True, in a rectangle diagonals bisect each other. 
 
(v) Its diagonals are perpendicular. 
False, Diagonals of a rectangle need not be perpendicular. 
 
(vi) Its diagonals are perpendicular and bisect each other. 
False, Diagonals of a rectangle need not be perpendicular. Diagonals only bisect each 
other. 
 
(vii) Its diagonals are equal and bisect each other. 
True, Diagonals are of equal length and bisect each other. 
 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
False, Diagonals are of equal length and bisect each other. Diagonals of a rectangle need 
not be perpendicular 
 
(ix) All rectangles are squares. 
False, in a square all sides are of equal length. 
 
(x) All rhombuses are parallelograms. 
True, all rhombuses are parallelograms, since opposite sides are equal and parallel. 
 
(xi) All squares are rhombuses and also rectangles. 
True, all squares are rhombuses, since all sides are equal in a square and rhombus. All 
squares are rectangles, since opposite sides are equal and parallel. 
 
(xii) All squares are not parallelograms. 
False, all squares are parallelograms, since opposite sides are parallel and equal. 
 
2. Which of the following statements are true for a square? 
(i) It is a rectangle. 
(ii) It has all its sides of equal length. 
(iii) Its diagonals bisect each other at right angle. 
(v) Its diagonals are equal to its sides. 
Solution: 
(i) It is a rectangle. 
True. Since, opposite sides are equal and parallel where, each angle is right angle. 
 
(ii) It has all its sides of equal length. 
True. Since, ides of a square are of equal length. 
 
(iii) Its diagonals bisect each other at right angle. 
True. Since, diagonals of a square bisect each other at right angle. 
 
(v) Its diagonals are equal to its sides. 
False. Since, diagonals of a square are of equal length. Length of diagonals is not equal to 
the length of sides 
 
3. Fill in the blanks in each of the following, so as to make the statement true : 
(i) A rectangle is a parallelogram in which ________. 
(ii) A square is a rhombus in which __________. 
(iii) A square is a rectangle in which ___________. 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
Solution: 
(i) A rectangle is a parallelogram in which one angle is a right angle. 
(ii) A square is a rhombus in which one angle is a right angle. 
(iii) A square is a rectangle in which adjacent sides are equal. 
 
4. A window frame has one diagonal longer than the other. Is the window frame a 
rectangle? Why or why not? 
Solution: 
No, diagonals of a rectangle are equal length. 
 
5. In a rectangle ABCD, prove that ?ACB ??CAD. 
Solution: 
Let us draw a rectangle, 
 
In rectangle ABCD, AC is the diagonal. 
In ?ACB and ?CAD 
AB = CD [Opposite sides of a rectangle are equal] 
BC = DA 
AC = CA [Common] 
By using SSS congruency 
?ACB ??CAD  
 
6. The sides of a rectangle are in the ratio 2 : 3, and its perimeter is 20 cm. Draw the 
rectangle. 
Solution: 
In rectangle ABCD, 
Given, perimeter of a rectangle = 20cm 
Ratio = 2:3 
So, let us consider the side as ‘x’ 
Length of rectangle (l) = 3x 
Breadth of the rectangle (b) = 2x 
We know that, 
Page 4


 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
EXERCISE 17.3                                                  PAGES NO: 17.22 
 
1. Which of the following statements are true for a rectangle? 
(i) It has two pairs of equal sides. 
(ii) It has all its sides of equal length. 
(iii) Its diagonals are equal. 
(iv) Its diagonals bisect each other. 
(v) Its diagonals are perpendicular. 
(vi) Its diagonals are perpendicular and bisect each other. 
(vii) Its diagonals are equal and bisect each other. 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
(ix) All rectangles are squares. 
(x) All rhombuses are parallelograms. 
(xi) All squares are rhombuses and also rectangles. 
(xii) All squares are not parallelograms. 
Solution: 
(i) It has two pairs of equal sides. 
 True, in a rectangle two pairs of sides are equal. 
 
(ii) It has all its sides of equal length. 
False, in a rectangle only two pairs of sides are equal. 
 
(iii) Its diagonals are equal. 
True, in a rectangle diagonals are of equal length. 
 
(iv) Its diagonals bisect each other. 
True, in a rectangle diagonals bisect each other. 
 
(v) Its diagonals are perpendicular. 
False, Diagonals of a rectangle need not be perpendicular. 
 
(vi) Its diagonals are perpendicular and bisect each other. 
False, Diagonals of a rectangle need not be perpendicular. Diagonals only bisect each 
other. 
 
(vii) Its diagonals are equal and bisect each other. 
True, Diagonals are of equal length and bisect each other. 
 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
False, Diagonals are of equal length and bisect each other. Diagonals of a rectangle need 
not be perpendicular 
 
(ix) All rectangles are squares. 
False, in a square all sides are of equal length. 
 
(x) All rhombuses are parallelograms. 
True, all rhombuses are parallelograms, since opposite sides are equal and parallel. 
 
(xi) All squares are rhombuses and also rectangles. 
True, all squares are rhombuses, since all sides are equal in a square and rhombus. All 
squares are rectangles, since opposite sides are equal and parallel. 
 
(xii) All squares are not parallelograms. 
False, all squares are parallelograms, since opposite sides are parallel and equal. 
 
2. Which of the following statements are true for a square? 
(i) It is a rectangle. 
(ii) It has all its sides of equal length. 
(iii) Its diagonals bisect each other at right angle. 
(v) Its diagonals are equal to its sides. 
Solution: 
(i) It is a rectangle. 
True. Since, opposite sides are equal and parallel where, each angle is right angle. 
 
(ii) It has all its sides of equal length. 
True. Since, ides of a square are of equal length. 
 
(iii) Its diagonals bisect each other at right angle. 
True. Since, diagonals of a square bisect each other at right angle. 
 
(v) Its diagonals are equal to its sides. 
False. Since, diagonals of a square are of equal length. Length of diagonals is not equal to 
the length of sides 
 
3. Fill in the blanks in each of the following, so as to make the statement true : 
(i) A rectangle is a parallelogram in which ________. 
(ii) A square is a rhombus in which __________. 
(iii) A square is a rectangle in which ___________. 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
Solution: 
(i) A rectangle is a parallelogram in which one angle is a right angle. 
(ii) A square is a rhombus in which one angle is a right angle. 
(iii) A square is a rectangle in which adjacent sides are equal. 
 
4. A window frame has one diagonal longer than the other. Is the window frame a 
rectangle? Why or why not? 
Solution: 
No, diagonals of a rectangle are equal length. 
 
5. In a rectangle ABCD, prove that ?ACB ??CAD. 
Solution: 
Let us draw a rectangle, 
 
In rectangle ABCD, AC is the diagonal. 
In ?ACB and ?CAD 
AB = CD [Opposite sides of a rectangle are equal] 
BC = DA 
AC = CA [Common] 
By using SSS congruency 
?ACB ??CAD  
 
6. The sides of a rectangle are in the ratio 2 : 3, and its perimeter is 20 cm. Draw the 
rectangle. 
Solution: 
In rectangle ABCD, 
Given, perimeter of a rectangle = 20cm 
Ratio = 2:3 
So, let us consider the side as ‘x’ 
Length of rectangle (l) = 3x 
Breadth of the rectangle (b) = 2x 
We know that, 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
Perimeter of the rectangle = 2(length + breadth) 
20 = 2(3x + 2x) 
10x = 20 
x = 20/10 = 2 
Length of the rectangle = 3×2 = 6cm 
Breadth of the rectangle = 2×2 = 4cm 
Here, is the diagram of rectangle 
 
 
7. The sides of a rectangle are in the ratio 4 : 5. Find its sides if the perimeter is 90 
cm. 
Solution: 
In rectangle ABCD, 
Given, perimeter of a rectangle = 90cm 
Ratio = 4:5 
So, let us consider the side as ‘x’ 
Length of rectangle (l) = 5x 
Breadth of the rectangle (b) = 4x 
We know that, 
Perimeter of the rectangle = 2(length + breadth) 
90 = 2(5x + 4x) 
18x = 90 
x = 90/18 = 5 
Length of the rectangle = 5×5 = 25cm 
Breadth of the rectangle = 4×5 = 20cm 
Here, is the diagram of rectangle 
 
Page 5


 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
EXERCISE 17.3                                                  PAGES NO: 17.22 
 
1. Which of the following statements are true for a rectangle? 
(i) It has two pairs of equal sides. 
(ii) It has all its sides of equal length. 
(iii) Its diagonals are equal. 
(iv) Its diagonals bisect each other. 
(v) Its diagonals are perpendicular. 
(vi) Its diagonals are perpendicular and bisect each other. 
(vii) Its diagonals are equal and bisect each other. 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
(ix) All rectangles are squares. 
(x) All rhombuses are parallelograms. 
(xi) All squares are rhombuses and also rectangles. 
(xii) All squares are not parallelograms. 
Solution: 
(i) It has two pairs of equal sides. 
 True, in a rectangle two pairs of sides are equal. 
 
(ii) It has all its sides of equal length. 
False, in a rectangle only two pairs of sides are equal. 
 
(iii) Its diagonals are equal. 
True, in a rectangle diagonals are of equal length. 
 
(iv) Its diagonals bisect each other. 
True, in a rectangle diagonals bisect each other. 
 
(v) Its diagonals are perpendicular. 
False, Diagonals of a rectangle need not be perpendicular. 
 
(vi) Its diagonals are perpendicular and bisect each other. 
False, Diagonals of a rectangle need not be perpendicular. Diagonals only bisect each 
other. 
 
(vii) Its diagonals are equal and bisect each other. 
True, Diagonals are of equal length and bisect each other. 
 
(viii) Its diagonals are equal and perpendicular, and bisect each other. 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
False, Diagonals are of equal length and bisect each other. Diagonals of a rectangle need 
not be perpendicular 
 
(ix) All rectangles are squares. 
False, in a square all sides are of equal length. 
 
(x) All rhombuses are parallelograms. 
True, all rhombuses are parallelograms, since opposite sides are equal and parallel. 
 
(xi) All squares are rhombuses and also rectangles. 
True, all squares are rhombuses, since all sides are equal in a square and rhombus. All 
squares are rectangles, since opposite sides are equal and parallel. 
 
(xii) All squares are not parallelograms. 
False, all squares are parallelograms, since opposite sides are parallel and equal. 
 
2. Which of the following statements are true for a square? 
(i) It is a rectangle. 
(ii) It has all its sides of equal length. 
(iii) Its diagonals bisect each other at right angle. 
(v) Its diagonals are equal to its sides. 
Solution: 
(i) It is a rectangle. 
True. Since, opposite sides are equal and parallel where, each angle is right angle. 
 
(ii) It has all its sides of equal length. 
True. Since, ides of a square are of equal length. 
 
(iii) Its diagonals bisect each other at right angle. 
True. Since, diagonals of a square bisect each other at right angle. 
 
(v) Its diagonals are equal to its sides. 
False. Since, diagonals of a square are of equal length. Length of diagonals is not equal to 
the length of sides 
 
3. Fill in the blanks in each of the following, so as to make the statement true : 
(i) A rectangle is a parallelogram in which ________. 
(ii) A square is a rhombus in which __________. 
(iii) A square is a rectangle in which ___________. 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
Solution: 
(i) A rectangle is a parallelogram in which one angle is a right angle. 
(ii) A square is a rhombus in which one angle is a right angle. 
(iii) A square is a rectangle in which adjacent sides are equal. 
 
4. A window frame has one diagonal longer than the other. Is the window frame a 
rectangle? Why or why not? 
Solution: 
No, diagonals of a rectangle are equal length. 
 
5. In a rectangle ABCD, prove that ?ACB ??CAD. 
Solution: 
Let us draw a rectangle, 
 
In rectangle ABCD, AC is the diagonal. 
In ?ACB and ?CAD 
AB = CD [Opposite sides of a rectangle are equal] 
BC = DA 
AC = CA [Common] 
By using SSS congruency 
?ACB ??CAD  
 
6. The sides of a rectangle are in the ratio 2 : 3, and its perimeter is 20 cm. Draw the 
rectangle. 
Solution: 
In rectangle ABCD, 
Given, perimeter of a rectangle = 20cm 
Ratio = 2:3 
So, let us consider the side as ‘x’ 
Length of rectangle (l) = 3x 
Breadth of the rectangle (b) = 2x 
We know that, 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
Perimeter of the rectangle = 2(length + breadth) 
20 = 2(3x + 2x) 
10x = 20 
x = 20/10 = 2 
Length of the rectangle = 3×2 = 6cm 
Breadth of the rectangle = 2×2 = 4cm 
Here, is the diagram of rectangle 
 
 
7. The sides of a rectangle are in the ratio 4 : 5. Find its sides if the perimeter is 90 
cm. 
Solution: 
In rectangle ABCD, 
Given, perimeter of a rectangle = 90cm 
Ratio = 4:5 
So, let us consider the side as ‘x’ 
Length of rectangle (l) = 5x 
Breadth of the rectangle (b) = 4x 
We know that, 
Perimeter of the rectangle = 2(length + breadth) 
90 = 2(5x + 4x) 
18x = 90 
x = 90/18 = 5 
Length of the rectangle = 5×5 = 25cm 
Breadth of the rectangle = 4×5 = 20cm 
Here, is the diagram of rectangle 
 
 
 
 
 
RD Sharma Solutions for Class 8 Maths Chapter 17 – 
Understanding Shapes – III (Special Types of Quadrilaterals) 
 
 
8. Find the length of the diagonal of a rectangle whose sides are 12 cm and 5 cm. 
Solution: 
 
 
In rectangle ABCD, 
Given, sides of a rectangle ABCD are 5cm and 12cm 
In ?ABC using Pythagoras theorem, 
AC
2
 = AB
2
 + BC
2
 
AC
2
 = 12
2
 + 5
2
 
AC
2
 = 144 + 25 
AC
2
 = 169 
AC = v169 
AC = 13cm 
? Length of the diagonal AC is 13cm. 
 
9. Draw a rectangle whose one side measures 8 cm and the length of each of whose 
diagonals is 10 cm. 
Solution: 
Given, one side of the rectangle is 8cm. 
Length of the diagonal = 10cm 
Now let us construct a rectangle, 
 
Steps to construct a rectangle, 
(i) Draw a line segment AB of length 8 cm 
(ii) From point ‘A’ cut an arc of length 10 cm and mark that point as C. 
(iii) From point B draw an angle of 90°, and join the arc from point A which cuts at point 
C. 
(iv) now join AC and BC 
(v) From point A draw an angle of 90° and from point C cut an arc of length 8 cm to get 
point D. 
(vi) Join CD and AD to form required rectangle.