Question 1

Which of the following is a correct definition of a rectangle?

Accepted Answer

A quadrilateral with four right angles and opposite sides equal and parallel — A rectangle is defined as a quadrilateral with four right angles (each 90°) and opposite sides that are equal and parallel. This distinguishes it from other quadrilaterals such as rhombus or trapezium.

Question 2

What is the special property that distinguishes a square from a rectangle?

Accepted Answer

All sides are equal and diagonals bisect each other at right angles — A square is a special type of rectangle where all four sides are equal in length and the diagonals bisect each other at right angles (90°), making them perpendicular. This property distinguishes squares from rectangles.

Question 3

In a rectangle ABCD, if AB = 7 cm and BC = 4 cm, what is the length of diagonal AC?

Accepted Answer

8.06 cm — Given: AB = 7 cm (length), BC = 4 cm (breadth)
Find: Length of diagonal AC
Formula: Diagonal of rectangle = \sqrt{l^2 + b^2}
Solution: Step 1: Substitute values: \sqrt{7^2 + 4^2} = \sqrt{49 + 16}
Step 2: Simplify: \sqrt{65}
Step 3: Calculate: 8.06 cm (approx)
Answer: 8.06 cm
Note: Students often forget to square both sides or use addition inside the square root.

Question 4

Fill in the blank: The formula to calculate the area of a rectangle is _____, where 'l' is length and 'b' is breadth.

Accepted Answer

l × b / length multiplied by breadth — The area of a rectangle is found by multiplying its length by its breadth, which gives the total number of square units enclosed within the rectangle.

Question 5

Explain why the diagonals of a rectangle are equal and bisect each other.

Accepted Answer

A rectangle has opposite sides equal and all angles are right angles. Because of this, the diagonals form two congruent triangles when drawn. These diagonals are equal in length and bisect each other at their point of intersection. For example, in rectangle ABCD, diagonals AC and BD are equal and intersect at point O, which divides both diagonals into equal halves. — The diagonals of a rectangle are equal because the rectangle consists of two congruent right-angled triangles formed by the diagonal. The bisecting property comes from the symmetry of the shape, where the diagonals cut each other into two equal parts.

Question 6

What is the perimeter of a square whose side length is 9 cm?

Accepted Answer

36 cm — Given: Side length a = 9 cm
Find: Perimeter P
Formula: P = 4 × a
Solution: Step 1: Substitute values: 4 × 9
Step 2: Calculate: 36 cm
Answer: 36 cm
Note: Students sometimes forget to multiply by 4 or confuse area with perimeter.

Question 7

True or False: The diagonals of a square are equal in length and bisect each other at right angles.

Accepted Answer

True — In a square, the diagonals are equal in length and they intersect at right angles (90°), bisecting each other. This is a unique property of squares that differentiates them from rectangles.

Question 8

Fill in the blank: The perimeter of a rectangle with length 'l' and breadth 'b' is given by the formula _____.

Accepted Answer

2(l + b) / 2 times (length plus breadth) — The perimeter of a rectangle is the total length around the rectangle, calculated by adding twice the length and twice the breadth, which is expressed as 2(l + b).

Question 9

Compare the properties of diagonals in a rectangle and a square. List at least two differences.

Accepted Answer

In a rectangle, the diagonals are equal in length and bisect each other but do not intersect at right angles. In a square, the diagonals are equal in length, bisect each other, and intersect at right angles (90°). Additionally, the diagonals of a square bisect the angles of the square, which is not true for a rectangle. — The key differences are:
1. Diagonals of a rectangle are equal and bisect each other but are not perpendicular.
2. Diagonals of a square are equal, bisect each other, and are perpendicular.
3. Diagonals of a square bisect the interior angles, unlike in a rectangle.

Question 10

A rectangle has a length of 12 m and a breadth of 7 m. Calculate its area and perimeter.

Accepted Answer

Area: 84 m², Perimeter: 38 m — Given: Length l = 12 m, Breadth b = 7 m
Find: Area and Perimeter
Formula: Area = l × b, Perimeter = 2(l + b)
Solution:
Step 1: Area = 12 × 7 = 84 m²
Step 2: Perimeter = 2(12 + 7) = 2 × 19 = 38 m
Answer: Area = 84 m², Perimeter = 38 m
Note: Students sometimes confuse area units with perimeter units.

Question 11

Why is the area of a square always calculated as side squared ($a^2$)? Explain with an example.

Accepted Answer

The area of a square is calculated as side squared because all sides are equal, so the number of unit squares inside the square is the side length multiplied by itself. For example, if the side length is 5 cm, the area is 5 × 5 = 25 cm², representing 25 unit squares covering the square. — Since a square has all sides equal, the length and breadth are the same, making the area formula $a 	imes a = a^2$. This formula directly counts the number of unit squares inside the square.

Question 12

A square and a rectangle both have a perimeter of 40 cm. The square has a side length of 10 cm. What is the length and breadth of the rectangle if its length is twice its breadth?

Accepted Answer

Length = 13.33 cm, Breadth = 6.67 cm — Given: Perimeter P = 40 cm, Length l = 2 × breadth b
Find: Length and breadth
Formula: P = 2(l + b)
Solution:
Step 1: Substitute l = 2b: 40 = 2(2b + b) = 2(3b) = 6b
Step 2: Solve for b: b = 40 / 6 = 6.67 cm
Step 3: Calculate l: l = 2 × 6.67 = 13.33 cm
Answer: Length = 13.33 cm, Breadth = 6.67 cm
Note: Students may forget to multiply breadth by 2 or divide perimeter correctly.

Question 13

True or False: In a rectangle, the diagonals always intersect at right angles.

Accepted Answer

False. In a rectangle, the diagonals are equal and bisect each other but do not intersect at right angles. — While the diagonals of a rectangle are equal in length and bisect each other, they do not intersect at 90°. This property is unique to squares and rhombuses.

Question 14

Match the following shapes with their correct properties:

Accepted Answer

— Matching properties helps understand differences and similarities between shapes.

Question 15

Calculate the area of a square whose diagonal measures 10 cm.

Accepted Answer

50 cm² — Given: Diagonal d = 10 cm
Find: Area of square
Formula: Side a = \frac{d}{\sqrt{2}}, Area = a^2
Solution:
Step 1: Calculate side: a = 10 / \sqrt{2} = 10 / 1.414 = 7.07 cm
Step 2: Calculate area: (7.07)^2 = 50 cm² (approx)
Answer: 50 cm²
Note: Students often forget to divide diagonal by \sqrt{2} to find side length.

AREA

AREA — Study Notes

Introduction

Properties of Rectangle

Properties of Square

Practice Questions — AREA

All 7 Chapters in Ganita Prakash Part-II