Question 1

The diagonals of a rhombus are 12 cm and 16 cm. Find the area of the rhombus.

Accepted Answer

Step 1: Recall the formula for area of a rhombus: A = (d₁ × d₂) / 2, where d₁ and d₂ are the diagonals. Step 2: Substitute d₁ = 12 cm and d₂ = 16 cm. Step 3: A = (12 × 16) / 2 = 192 / 2 = 96 cm². Therefore, the area is 96 cm².

Question 2

A rectangle has length 24 cm and breadth 16 cm. What is the perimeter of the rectangle?

Accepted Answer

Step 1: Recall the formula for perimeter of a rectangle: P = 2(l + b). Step 2: Substitute l = 24 cm and b = 16 cm. Step 3: P = 2(24 + 16) = 2(40) = 80 cm. Therefore, the perimeter is 80 cm.

Question 3

A parallelogram has a base of 20 cm and a height of 15 cm. What is its area?

Accepted Answer

Step 1: Recall the formula for area of a parallelogram: A = base × height. Step 2: Substitute base = 20 cm and height = 15 cm. Step 3: A = 20 × 15 = 300 cm². Therefore, the area is 300 cm².

Question 4

A trapezium has parallel sides of 10 cm and 14 cm, and a height of 8 cm. Calculate its area.

Accepted Answer

Step 1: Recall the formula for area of a trapezium: A = [(a + b) / 2] × h, where a and b are the parallel sides and h is the height. Step 2: Substitute a = 10 cm, b = 14 cm, and h = 8 cm. Step 3: A = [(10 + 14) / 2] × 8 = [24 / 2] × 8 = 12 × 8 = 96 cm². Therefore, the area is 96 cm².

Question 5

A square has a side length of 9 cm. What is the length of its diagonal?

Accepted Answer

Step 1: Recall that for a square with side a, the diagonal d = a√2. Step 2: Substitute a = 9 cm. Step 3: d = 9√2 cm. Step 4: Calculating the numerical value: d = 9 × 1.414... ≈ 12.73 cm. However, if the question expects the exact form, the answer is 9√2 cm. For this exam context, 9√2 cm is the standard answer.

Question 6

A rectangle has length 30 cm and diagonal 34 cm. Find the breadth of the rectangle.

Accepted Answer

Step 1: In a rectangle, the diagonal, length, and breadth form a right triangle. Use the Pythagorean theorem: diagonal² = length² + breadth². Step 2: Substitute diagonal = 34 cm and length = 30 cm. Step 3: 34² = 30² + b². Step 4: 1156 = 900 + b². Step 5: b² = 1156 - 900 = 256. Step 6: b = √256 = 16 cm. Therefore, the breadth is 16 cm.

Question 7

A parallelogram has adjacent sides of length 12 cm and 8 cm. If the angle between them is 60°, what is the area of the parallelogram?

Accepted Answer

Step 1: For a parallelogram, Area = base × height = a × b × sin(θ), where a and b are adjacent sides and θ is the included angle. Step 2: Area = 12 × 8 × sin(60°) = 12 × 8 × (√3/2) = 96 × (√3/2) = 48√3 ≈ 83.14 cm². Step 3: Rounding to nearest practical value gives 48√3 cm².

Question 8

A rhombus has diagonals of length 16 cm and 12 cm. What is the perimeter of the rhombus?

Accepted Answer

Step 1: In a rhombus, diagonals bisect each other at right angles. Step 2: Half-diagonals are 8 cm and 6 cm. Step 3: Each side of the rhombus forms the hypotenuse of a right triangle with legs 8 cm and 6 cm. Step 4: Side = √(8² + 6²) = √(64 + 36) = √100 = 10 cm. Step 5: Perimeter = 4 × 10 = 40 cm.

Question 9

A rectangle has length 20 cm and width 15 cm. If each corner is cut off by a right-angled isosceles triangle with legs of 3 cm, what is the perimeter of the resulting octagon?

Accepted Answer

Step 1: Original rectangle: length = 20 cm, width = 15 cm. Step 2: Each corner cut by isosceles right triangle with legs = 3 cm. Step 3: Hypotenuse of each triangle = 3√2 cm. Step 4: After cutting all 4 corners, the octagon has: 4 sides from the original rectangle edges (reduced) and 4 hypotenuses. Step 5: The two longer sides (20 cm each) lose 3 cm from each end: 20 - 2(3) = 14 cm each. Step 6: The two shorter sides (15 cm each) lose 3 cm from each end: 15 - 2(3) = 9 cm each. Step 7: Perimeter = 2(14) + 2(9) + 4(3√2) = 28 + 18 + 12√2 = 46 + 12√2 cm.

Question 10

A quadrilateral ABCD is a cyclic quadrilateral inscribed in a circle. If angles A and C are opposite angles, and angle A = 75°, what is angle C?

Accepted Answer

Step 1: In a cyclic quadrilateral, opposite angles are supplementary (sum to 180°). Step 2: Angle A + Angle C = 180°. Step 3: 75° + Angle C = 180°. Step 4: Angle C = 180° - 75° = 105°.

Question 11

A kite has two pairs of adjacent equal sides. If one pair of opposite angles are 60° and 120°, and the kite is divided by its diagonals into four triangles, what is the measure of one of the other two opposite angles if the sum of all angles in the kite is 360°?

Accepted Answer

Step 1: A kite has 4 angles that sum to 360°. Step 2: Given two opposite angles: 60° and 120°. Step 3: Sum of these two = 60° + 120° = 180°. Step 4: The remaining two opposite angles must sum to 360° - 180° = 180°. Step 5: In a kite, one pair of opposite angles (the ones between unequal sides) are equal. Step 6: Therefore, each of the remaining two opposite angles = 180° / 2 = 90°.

Question 12

In a parallelogram PQRS, the diagonals PR and QS intersect at point O. If PO = 8 cm, QO = 6 cm, and the angle between the diagonals is 60°, find the area of the parallelogram (in cm²).

Accepted Answer

Step 1: In a parallelogram, diagonals bisect each other. So PR = 2 × PO = 16 cm and QS = 2 × QO = 12 cm. Step 2: The area of a parallelogram can be calculated using diagonals: Area = (1/2) × d₁ × d₂ × sin(θ), where d₁ and d₂ are diagonals and θ is the angle between them. Step 3: Area = (1/2) × 16 × 12 × sin(60°) = (1/2) × 16 × 12 × (√3/2) = 96 × (√3/2) = 48√3 cm². Step 4: 48√3 ≈ 48 × 1.732 ≈ 83.14 cm². For clean answer in multiple choice: 48√3 ≈ 83 cm² or exact form.

SBI PO Quadrilaterals

Quadrilaterals— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Quadrilaterals — Practice Questions

60-Second Revision — Quadrilaterals