Question 1

An equilateral triangle has a side length of 10 cm. What is its perimeter?

Accepted Answer

Step 1: An equilateral triangle has all three sides equal. Step 2: Perimeter = 3 × side length = 3 × 10 = 30 cm. Therefore, the perimeter is 30 cm.

Question 2

A right-angled triangle has legs of length 6 cm and 8 cm. What is the length of its hypotenuse?

Accepted Answer

Step 1: Use the Pythagorean theorem: c² = a² + b², where c is the hypotenuse and a, b are the legs. Step 2: c² = 6² + 8² = 36 + 64 = 100. Step 3: c = √100 = 10 cm. Therefore, the hypotenuse is 10 cm.

Question 3

A triangle has sides of length 5 cm, 12 cm, and 13 cm. What is its area using Heron's formula?

Accepted Answer

Step 1: Calculate the semi-perimeter: s = (5 + 12 + 13) ÷ 2 = 30 ÷ 2 = 15 cm. Step 2: Apply Heron's formula: Area = √[s(s−a)(s−b)(s−c)] = √[15 × 10 × 3 × 2] = √900 = 30 cm². Therefore, the area is 30 cm².

Question 4

The area of a triangle is 24 cm² and its base is 6 cm. What is its height?

Accepted Answer

Step 1: Use the area formula rearranged for height: Area = (1/2) × base × height, so height = (2 × Area) ÷ base. Step 2: height = (2 × 24) ÷ 6 = 48 ÷ 6 = 8 cm. Therefore, the height is 8 cm.

Question 5

A triangle has a base of 12 cm and a height of 8 cm. What is its area?

Accepted Answer

Step 1: Use the formula for the area of a triangle: Area = (1/2) × base × height. Step 2: Substitute values: Area = (1/2) × 12 × 8 = (1/2) × 96 = 48 cm². Therefore, the area is 48 cm².

Question 6

A triangle has an area of 60 cm² and a base of 15 cm. If the base is increased by 20% and the height is decreased by 20%, what is the new area?

Accepted Answer

Step 1: Original area = 60 cm², base = 15 cm. Using area = (1/2) × base × height: 60 = (1/2) × 15 × h, so h = 8 cm. Step 2: New base = 15 × 1.2 = 18 cm. New height = 8 × 0.8 = 6.4 cm. Step 3: New area = (1/2) × 18 × 6.4 = (1/2) × 115.2 = 57.6 cm². Therefore, the new area is 57.6 cm².

Question 7

In triangle ABC, the sides are in the ratio 3 : 4 : 5. If the perimeter is 72 cm, what is the area of the triangle?

Accepted Answer

Step 1: Let sides be 3x, 4x, 5x. Perimeter = 3x + 4x + 5x = 12x = 72, so x = 6. Step 2: Sides are 18 cm, 24 cm, 30 cm. Step 3: Check if right triangle: 18² + 24² = 324 + 576 = 900 = 30². Yes, it's a right triangle. Step 4: Area = (1/2) × 18 × 24 = (1/2) × 432 = 216 cm². Therefore, the area is 216 cm².

Question 8

A triangle has vertices at coordinates A(0, 0), B(8, 0), and C(4, 6). What is its area?

Accepted Answer

Step 1: Use the coordinate geometry formula: Area = (1/2)|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|. Step 2: Substitute A(0,0), B(8,0), C(4,6): Area = (1/2)|0(0 - 6) + 8(6 - 0) + 4(0 - 0)| = (1/2)|0 + 48 + 0| = (1/2) × 48 = 24 cm². Therefore, the area is 24 cm².

Question 9

A triangle has sides of length 13 cm, 14 cm, and 15 cm. What is its area?

Accepted Answer

Step 1: Use Heron's formula. Semi-perimeter s = (13 + 14 + 15)/2 = 21 cm. Step 2: Area = √[s(s-a)(s-b)(s-c)] = √[21 × 8 × 7 × 6] = √7056 = 84 cm². Therefore, the area is 84 cm².

Question 10

A triangle has sides 13 cm, 14 cm, and 15 cm. An altitude is drawn from the vertex opposite the 14 cm side to that side. What is the length of this altitude in cm?

Accepted Answer

Step 1: Use Heron's formula to find the area. Semi-perimeter s = (13 + 14 + 15)/2 = 21 cm. Step 2: Area = √[21(21-13)(21-14)(21-15)] = √[21 × 8 × 7 × 6] = √7056 = 84 cm². Step 3: Using Area = (1/2) × base × height, we have 84 = (1/2) × 14 × h, so h = 12 cm.

Question 11

In triangle PQR, the sides are PQ = 10 cm, QR = 17 cm, and PR = 21 cm. What is the area of the triangle in cm²?

Accepted Answer

Step 1: Calculate semi-perimeter s = (10 + 17 + 21)/2 = 48/2 = 24 cm. Step 2: Apply Heron's formula: Area = √[s(s-a)(s-b)(s-c)] = √[24(24-10)(24-17)(24-21)] = √[24 × 14 × 7 × 3] = √7056 = 84 cm².

Question 12

A right-angled triangle has legs of length 9 cm and 12 cm. A circle is inscribed in this triangle. What is the radius of the inscribed circle in cm?

Accepted Answer

Step 1: The hypotenuse c = √(9² + 12²) = √(81 + 144) = √225 = 15 cm. Step 2: For a right triangle, the inradius r = (a + b - c)/2, where a and b are legs and c is the hypotenuse. Step 3: r = (9 + 12 - 15)/2 = 6/2 = 3 cm.

SBI Clerk Triangles — Area & Properties

Triangles — Area & Properties— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Triangles — Area & Properties — Practice Questions

60-Second Revision — Triangles — Area & Properties