Question 1

A triangle has sides of length 5 cm, 12 cm, and 13 cm. What is its semi-perimeter?

Accepted Answer

Step 1: Calculate the perimeter by adding all three sides: Perimeter = 5 + 12 + 13 = 30 cm. Step 2: Semi-perimeter (s) = Perimeter ÷ 2 = 30 ÷ 2 = 15 cm. Therefore, the semi-perimeter is 15 cm.

Question 2

An isosceles triangle has two equal sides of 13 cm each and a base of 10 cm. What is its perimeter?

Accepted Answer

Step 1: Identify the three sides: two equal sides of 13 cm and one base of 10 cm. Step 2: Perimeter = 13 + 13 + 10 = 36 cm. Therefore, the perimeter is 36 cm.

Question 3

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 = 3 × 10 = 30 cm. Therefore, the perimeter is 30 cm.

Question 4

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. Step 2: Substitute values: c² = 6² + 8² = 36 + 64 = 100. Step 3: Take the square root: c = √100 = 10 cm. Therefore, the hypotenuse is 10 cm.

Question 5

The area of a triangle is 54 cm² and its base is 12 cm. What is the height of the triangle?

Accepted Answer

Step 1: Use the triangle area formula: Area = (1/2) × base × height. Step 2: Rearrange to find height: height = (2 × Area) ÷ base. Step 3: Substitute values: height = (2 × 54) ÷ 12 = 108 ÷ 12 = 9 cm. Therefore, the height is 9 cm.

Question 6

In triangle ABC, angle B = 90°, AB = 5 cm, and BC = 12 cm. The angle bisector from B meets AC at point D. What is the length of BD in cm?

Accepted Answer

Step 1: Since angle B = 90°, the angle bisector from B divides the right angle into two 45° angles. Step 2: AC = √(5² + 12²) = 13 cm. Step 3: In a right triangle, the angle bisector from the right angle to the hypotenuse has length: BD = (AB × BC) / (AB + BC) × √2 = (5 × 12) / (5 + 12) × √2 = (60/17) × √2 ≈ 4.99 cm. Alternatively, using the formula for angle bisector length: BD = (2 × AB × BC × cos(45°)) / (AB + BC) = (2 × 5 × 12 × (1/√2)) / 17 = (120/√2) / 17 = (60√2) / 17 ≈ 4.99 cm. Step 4: More directly: BD = (AB × BC × √2) / (AB + BC) = (5 × 12 × √2) / 17 = (60√2) / 17 cm ≈ 4.99 cm, which simplifies to approximately 5 cm or exactly (60√2)/17 cm.

Question 7

A triangle has sides of length 13 cm, 14 cm, and 15 cm. What is its area in cm²?

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.

Question 8

An equilateral triangle has a side length of 10 cm. A smaller equilateral triangle is inscribed inside it such that its vertices touch the midpoints of the sides of the larger triangle. What is the ratio of the area of the smaller triangle to the area of the larger triangle?

Accepted Answer

Step 1: When vertices of an inscribed equilateral triangle touch the midpoints of the sides of a larger equilateral triangle, the side of the smaller triangle is half the side of the larger. Step 2: Side of smaller triangle = 10 / 2 = 5 cm. Step 3: Area ratio = (side ratio)² = (1/2)² = 1/4.

Question 9

A triangle has vertices at coordinates A(0, 0), B(8, 0), and C(4, 6). What is the area of this triangle in square units?

Accepted Answer

Step 1: Use the coordinate formula: Area = (1/2) |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|. Step 2: Area = (1/2) |0(0 - 6) + 8(6 - 0) + 4(0 - 0)| = (1/2) |0 + 48 + 0| = (1/2) × 48 = 24 square units.

Question 10

In triangle ABC, the sides are in the ratio 5:6:7. If the perimeter is 72 cm, what is the area of the triangle (in cm²)?

Accepted Answer

Step 1: Sides are 5k, 6k, 7k with perimeter 18k = 72, so k = 4. Sides are 20, 24, 28 cm. Step 2: s = 36 cm. Step 3: Area = √[36 × 16 × 12 × 8] = √55296 = 96√6 cm² ≈ 235.2 cm².

Question 11

A triangle has sides 13 cm, 14 cm, and 15 cm. A perpendicular is drawn from the vertex opposite the 14 cm side to that side. What is the length of this perpendicular (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. Step 4: h = (84 × 2)/14 = 168/14 = 12 cm.

SBI PO 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