Question 1

A parallelogram has a base of 24 cm and a height of 15 cm. If the base is increased by 8 cm while the height remains the same, by what percentage does the area increase?

Accepted Answer

Step 1: Original area of parallelogram = base × height = 24 × 15 = 360 cm². Step 2: New base = 24 + 8 = 32 cm. New area = 32 × 15 = 480 cm². Step 3: Increase in area = 480 - 360 = 120 cm². Step 4: Percentage increase = (120 / 360) × 100 = 33.33% or 33⅓%.

Question 2

A parallelogram has a base of 24 cm and a height of 15 cm. If one of its sides is 20 cm, what is the perimeter of the parallelogram?

Accepted Answer

Step 1: In a parallelogram, opposite sides are equal. So if one side is 24 cm (base), the opposite side is also 24 cm. Similarly, if another side is 20 cm, the opposite side is also 20 cm. Step 2: Perimeter = 2(base + side) = 2(24 + 20) = 2(44) = 88 cm. Note: The height is given but not needed to find the perimeter—it would be used only if area was asked. Therefore, the perimeter is 88 cm.

Question 3

A trapezium has parallel sides of 14 cm and 10 cm, and height 6 cm. Find its area.

Accepted Answer

Step 1: Recall the formula for area of a trapezium: A = (1/2) × (sum of parallel sides) × height. Step 2: Substitute the parallel sides as 14 cm and 10 cm, and height = 6 cm. Step 3: A = (1/2) × (14 + 10) × 6 = (1/2) × 24 × 6 = (1/2) × 144 = 72 cm². Therefore, the area is 72 cm².

Question 4

A parallelogram has base 18 cm and height 10 cm. What is its area?

Accepted Answer

Step 1: Recall the formula for area of a parallelogram: A = base × height. Step 2: Substitute base = 18 cm and height = 10 cm. Step 3: A = 18 × 10 = 180 cm². Therefore, the area is 180 cm².

Question 5

In a rectangle ABCD, the length is 20 cm and the breadth is 15 cm. A diagonal AC is drawn. If a point P is taken on diagonal AC such that AP:PC = 2:3, what is the perpendicular distance from point P to side AB?

Accepted Answer

Step 1: Set up coordinates with A at origin (0, 0), B at (20, 0), C at (20, 15), and D at (0, 15). Step 2: Diagonal AC goes from (0, 0) to (20, 15). Step 3: Point P divides AC in ratio 2:3, so P = A + (2/5)(C − A) = (0, 0) + (2/5)(20, 15) = (8, 6). Step 4: The perpendicular distance from P to side AB (which lies on the x-axis, y = 0) is simply the y-coordinate of P, which is 6 cm.

Question 6

In a kite PQRS, the diagonals PR and QS intersect at O. If PR = 20 cm, QS = 16 cm, and the diagonals intersect such that PO = 12 cm and QO = 6 cm, find the area of the kite.

Accepted Answer

Step 1: The area of a kite is given by: Area = (1/2) × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals. Step 2: The diagonals of a kite are perpendicular to each other. Step 3: Area = (1/2) × 20 × 16 = (1/2) × 320 = 160 cm². Note: The specific intersection points (PO = 12, QO = 6) confirm the kite properties but do not change the diagonal lengths used in the formula.

Question 7

A quadrilateral has vertices at A(0, 0), B(6, 0), C(6, 4), and D(0, 4). What is the area of quadrilateral ABCD?

Accepted Answer

Step 1: Plot the points: A(0,0), B(6,0), C(6,4), D(0,4). Step 2: Observe that AB is horizontal (length 6), BC is vertical (length 4), CD is horizontal (length 6), and DA is vertical (length 4). Step 3: This forms a rectangle with length 6 and width 4. Step 4: Area = length × width = 6 × 4 = 24 cm².

Question 8

In a rectangle, the length is 5 cm more than the width. If the perimeter is 50 cm, find the area of the rectangle.

Accepted Answer

Step 1: Let width = w cm, then length = (w + 5) cm. Step 2: Perimeter = 2(length + width) = 2(w + 5 + w) = 2(2w + 5) = 4w + 10 = 50. Step 3: 4w = 40, so w = 10 cm. Step 4: Length = 10 + 5 = 15 cm. Step 5: Area = length × width = 15 × 10 = 150 cm².

Question 9

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, the 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 10

The diagonals of a rhombus are 24 cm and 10 cm. What is the perimeter of the rhombus?

Accepted Answer

Step 1: In a rhombus, diagonals bisect each other at right angles. So half-diagonals are 12 cm and 5 cm. Step 2: Each side of the rhombus = √(12² + 5²) = √(144 + 25) = √169 = 13 cm. Step 3: Perimeter = 4 × 13 = 52 cm. Therefore, answer is 52 cm.

Question 11

A parallelogram has a base of 24 cm and a height of 15 cm. If the base is increased by 25% and the height is decreased by 20%, what is the percentage change in the area of the parallelogram?

Accepted Answer

Step 1: Original area = base × height = 24 × 15 = 360 cm². Step 2: New base = 24 + (25% of 24) = 24 + 6 = 30 cm. Step 3: New height = 15 − (20% of 15) = 15 − 3 = 12 cm. Step 4: New area = 30 × 12 = 360 cm². Step 5: Percentage change = [(360 − 360) / 360] × 100 = 0%. Therefore, the area remains unchanged.

Question 12

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

Accepted Answer

Step 1: Use the formula for area of 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 cm². Step 3: 48√3 ≈ 48 × 1.732 = 83.14 cm². However, for exact answer: 48√3 cm².

Question 13

A quadrilateral ABCD has diagonals AC and BD intersecting at point O. If AC = 24 cm, BD = 18 cm, and the diagonals intersect at right angles, find the area of the quadrilateral. The diagonals divide the quadrilateral such that AO = 10 cm, OC = 14 cm, BO = 8 cm, and OD = 10 cm.

Accepted Answer

Step 1: When diagonals of a quadrilateral intersect at right angles, the area formula is (1/2) × d₁ × d₂, where d₁ and d₂ are the full diagonal lengths. Step 2: Area = (1/2) × 24 × 18 = (1/2) × 432 = 216 cm². The individual segment lengths (AO, OC, BO, OD) are provided as distractors but are not needed for the standard formula when diagonals are perpendicular.

Question 14

A rectangle PQRS has length 24 cm and width 10 cm. A diagonal PR is drawn. A circle is drawn such that it passes through points P, Q, and R (i.e., it is the circumcircle of triangle PQR). Find the radius of this circle.

Accepted Answer

Step 1: The circumcircle of triangle PQR passes through P, Q, R. Since PQRS is a rectangle, angle PQR = 90°. Step 2: By Thales' theorem, the hypotenuse PR is the diameter of the circumcircle. Step 3: PR = √(24² + 10²) = √676 = 26 cm. Step 4: Radius = 26/2 = 13 cm.

Question 15

A trapezium has parallel sides of lengths 14 cm and 10 cm, and the perpendicular distance between them is 8 cm. Find the area of the trapezium.

Accepted Answer

Step 1: Recall the formula for area of a trapezium: A = (1/2) × (sum of parallel sides) × height. Step 2: Substitute the parallel sides as 14 cm and 10 cm, and height = 8 cm. Step 3: A = (1/2) × (14 + 10) × 8 = (1/2) × 24 × 8 = (1/2) × 192 = 96 cm². Therefore, the area is 96 cm².

Question 16

A rectangle has length 20 cm and diagonal 25 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 = 25 cm and length = 20 cm. Step 3: 25² = 20² + b². Step 4: 625 = 400 + b². Step 5: b² = 225. Step 6: b = 15 cm. Therefore, the breadth is 15 cm.

Question 17

A rhombus has diagonals of length 20 cm and 15 cm. What is its area?

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 = 20 cm and d₂ = 15 cm. Step 3: A = (20 × 15) / 2 = 300 / 2 = 150 cm². Therefore, the area is 150 cm².

Question 18

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 19

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

Accepted Answer

Step 1: Recall that in a square with side length a, the diagonal d = a√2. Step 2: Substitute a = 9 cm. Step 3: d = 9√2 cm. Step 4: Calculating numerically: d = 9 × 1.414... ≈ 12.73 cm. However, the exact answer is 9√2 cm. If numerical form is required: approximately 12.73 cm, but the standard answer is 9√2 cm.

Question 20

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

Question 21

A trapezium has parallel sides of 12 cm and 18 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 = 12 cm, b = 18 cm, and h = 8 cm. Step 3: A = [(12 + 18) / 2] × 8 = [30 / 2] × 8 = 15 × 8 = 120 cm². Therefore, the area is 120 cm².

Question 22

A trapezium has parallel sides of 14 cm and 10 cm, and a height of 6 cm. What is the area of the trapezium?

Accepted Answer

Step 1: Use 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 = 14 cm, b = 10 cm, and h = 6 cm. Step 3: A = [(14 + 10) / 2] × 6 = [24 / 2] × 6 = 12 × 6 = 72 cm². Therefore, the area is 72 cm².

Question 23

The diagonals of a rhombus are 12 cm and 16 cm. What is the area of the rhombus?

Accepted Answer

Step 1: Use 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 24

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

Accepted Answer

Step 1: Use 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 25

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

Question 26

A parallelogram has a base of 18 cm and a height of 10 cm. What is its area?

Accepted Answer

Step 1: Use the formula for area of a parallelogram: A = base × height. Step 2: Substitute base = 18 cm and height = 10 cm. Step 3: A = 18 × 10 = 180 cm². Therefore, the area is 180 cm².

Question 27

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

Accepted Answer

Step 1: Use the formula for diagonal of a square: d = side × √2. Step 2: Substitute side = 9 cm. Step 3: d = 9√2 cm. Step 4: Calculating the numerical value: d = 9 × 1.414... ≈ 12.73 cm. However, the exact answer is 9√2 cm. If we need a numerical approximation from the options, 9√2 ≈ 12.7 cm, which rounds to approximately 12.7 or 13 cm depending on rounding. For SSC purposes, the answer is 9√2 cm.

Question 28

A parallelogram has base 18 cm and height 12 cm. Find its area.

Accepted Answer

Step 1: Recall the formula for area of a parallelogram: A = base × height. Step 2: Substitute base = 18 cm and height = 12 cm. Step 3: A = 18 × 12 = 216 cm². Therefore, the area is 216 cm².

Question 29

A rectangle has length 20 cm and diagonal 25 cm. What is the breadth of the rectangle?

Accepted Answer

Step 1: Use the Pythagorean theorem for a rectangle: diagonal² = length² + breadth². Step 2: Substitute diagonal = 25 cm and length = 20 cm. Step 3: 25² = 20² + breadth². Step 4: 625 = 400 + breadth². Step 5: breadth² = 625 - 400 = 225. Step 6: breadth = √225 = 15 cm. Therefore, the breadth is 15 cm.

Question 30

A parallelogram has base 18 cm and height 10 cm. What is its area?

Accepted Answer

Step 1: Recall the formula for area of a parallelogram: A = base × height. Step 2: Substitute base = 18 cm and height = 10 cm. Step 3: A = 18 × 10 = 180 cm². Therefore, the area is 180 cm².

Question 31

A parallelogram has a base of 24 cm and a height of 15 cm. If the base is increased by 8 cm while the height remains the same, by what percentage does the area increase?

Accepted Answer

Step 1: Original area of parallelogram = base × height = 24 × 15 = 360 cm². Step 2: New base = 24 + 8 = 32 cm. New area = 32 × 15 = 480 cm². Step 3: Increase in area = 480 − 360 = 120 cm². Step 4: Percentage increase = (120 ÷ 360) × 100 = 33.33% or 33⅓%. Therefore, the answer is 33⅓%.

Question 32

In a cyclic quadrilateral ABCD, the sides are AB = 6 cm, BC = 8 cm, CD = 7 cm, and DA = 5 cm. Using Brahmagupta's formula, what is the area of the quadrilateral (approximately)?

Accepted Answer

Step 1: For a cyclic quadrilateral, use Brahmagupta's formula: Area = √[(s-a)(s-b)(s-c)(s-d)], where s is the semi-perimeter. Step 2: s = (6 + 8 + 7 + 5)/2 = 26/2 = 13 cm. Step 3: (s - a) = 13 - 6 = 7, (s - b) = 13 - 8 = 5, (s - c) = 13 - 7 = 6, (s - d) = 13 - 5 = 8. Step 4: Area = √[7 × 5 × 6 × 8] = √[1680] = √[16 × 105] = 4√105 ≈ 4 × 10.247 ≈ 40.99 ≈ 41 cm².

Question 33

In a cyclic quadrilateral PQRS, the sides are PQ = 6 cm, QR = 8 cm, RS = 7 cm, and SP = 5 cm. Using Brahmagupta's formula, find the area of the quadrilateral (in cm²).

Accepted Answer

Step 1: For a cyclic quadrilateral, use Brahmagupta's formula: Area = √[(s-a)(s-b)(s-c)(s-d)], where s is the semi-perimeter. Step 2: s = (6 + 8 + 7 + 5)/2 = 26/2 = 13 cm. Step 3: Area = √[(13-6)(13-8)(13-7)(13-5)] = √[7 × 5 × 6 × 8] = √1680. Step 4: √1680 = √(16 × 105) = 4√105 ≈ 41 cm² (or exact: 4√105).

Question 34

A trapezium ABCD has parallel sides AB and CD of lengths 18 cm and 12 cm respectively. The perpendicular distance between these parallel sides is 10 cm. If a line parallel to both AB and CD divides the trapezium into two smaller trapeziums of equal area, at what distance from AB does this dividing line lie?

Accepted Answer

Step 1: Area of trapezium ABCD = (1/2) × (18 + 12) × 10 = 150 cm². Step 2: Each smaller trapezium must have area 75 cm². Step 3: Let the dividing line be at distance h from AB with length x. The upper trapezium has parallel sides 18 and x with height h. Area = (1/2) × (18 + x) × h = 75. Step 4: By the property of trapeziums divided by a line parallel to the bases, if the dividing line is at distance h from AB, then x = √[(18² + 12²)/2] = √[(324 + 144)/2] = √234 ≈ but we use exact formula: x = √(18 × 12) = √216 = 6√6 cm (geometric mean for equal area division). Step 5: Using (1/2)(18 + 6√6) × h = 75, we get h = 150/(18 + 6√6). Rationalizing: h = 150(18 - 6√6)/[(18 + 6√6)(18 - 6√6)] = 150(18 - 6√6)/(324 - 216) = 150(18 - 6√6)/108 = 25(18 - 6√6)/18. Simplifying: h = (450 - 150√6)/18 = 25 - (25√6)/3. Alternatively, using direct formula: h² = (h₁² + h₂²)/2 where h₁ = 10, so h = 5√2 cm ≈ 7.07 cm. Recalculating: For equal areas with parallel sides a=18, b=12, height H=10, dividing line at height h has length c where (18+c)h = (c+12)(10-h). Solving: h = 10(18+12)/(18+12+2√(18×12)) = 300/(30+2×6√6) is complex. Using standard result: h = 5√2 cm.

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