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

An isosceles triangle has two equal sides of 7 cm each and a base of 10 cm. What is its perimeter?

Accepted Answer

Step 1: An isosceles triangle has two sides of equal length. Step 2: Perimeter = side₁ + side₂ + base = 7 + 7 + 10 = 24 cm. Therefore, the perimeter is 24 cm.

Question 3

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

Accepted Answer

Step 1: Use the area formula: Area = (1/2) × base × height. Step 2: Rearrange to find height: height = (2 × Area) ÷ base. Step 3: height = (2 × 24) ÷ 6 = 48 ÷ 6 = 8 cm. Therefore, the height is 8 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, 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 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 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 sides of length 5 cm, 12 cm, and 13 cm. What is its semi-perimeter?

Accepted Answer

Step 1: Calculate the perimeter: P = 5 + 12 + 13 = 30 cm. Step 2: Semi-perimeter (s) = P ÷ 2 = 30 ÷ 2 = 15 cm. Therefore, the semi-perimeter is 15 cm.

Question 7

An equilateral triangle has a side length of 8 cm. What is the ratio of its area to the area of a square with the same perimeter?

Accepted Answer

Step 1: Perimeter of equilateral triangle = 3 × 8 = 24 cm. Step 2: Area of equilateral triangle = (√3/4) × 8² = (√3/4) × 64 = 16√3 cm². Step 3: Square with perimeter 24 cm has side = 24/4 = 6 cm. Step 4: Area of square = 6² = 36 cm². Step 5: Ratio = 16√3 : 36 = 4√3 : 9. Therefore, the ratio is 4√3 : 9.

Question 8

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 9

A triangle has vertices at A(0, 0), B(8, 0), and C(4, 6). Find the area of the triangle using the coordinate formula.

Accepted Answer

Step 1: Use the coordinate area 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 10

A triangle has sides of length 13 cm, 14 cm, and 15 cm. Using Heron's formula, find its area.

Accepted Answer

Step 1: Calculate semi-perimeter s = (13 + 14 + 15)/2 = 42/2 = 21 cm. Step 2: Apply Heron's formula: Area = √[s(s-a)(s-b)(s-c)] = √[21 × 8 × 7 × 6] = √7056 = 84 cm². Therefore, the area is 84 cm².

Question 11

In an isosceles triangle, the base is 16 cm and each of the equal sides is 10 cm. A line parallel to the base divides the triangle into two parts of equal area. Find the distance of this parallel line from the base (in cm).

Accepted Answer

Step 1: Height of isosceles triangle = √(10² - 8²) = 6 cm. Step 2: Original area = 48 cm². Step 3: Smaller similar triangle has area 24 cm². Step 4: Linear scale factor = 1/√2. Step 5: Height of smaller triangle = 6/√2 = 3√2 cm. Step 6: Distance from base = 6 - 3√2 = 3(2 - √2) cm ≈ 1.76 cm.

Question 12

In triangle PQR, the sides are PQ = 25 cm, QR = 29 cm, and PR = 36 cm. What is the area of the triangle (in cm²)?

Accepted Answer

Step 1: Semi-perimeter s = (25 + 29 + 36)/2 = 90/2 = 45 cm. Step 2: Area = √[s(s-PQ)(s-QR)(s-PR)] = √[45 × 20 × 16 × 9]. Step 3: Rearrange for easy calculation: (45 × 20) × (16 × 9) = 900 × 144 = 129,600. Step 4: √129,600 = √(900 × 144) = 30 × 12 = 360 cm².

Question 13

Two triangles have the same height. The base of the first triangle is 12 cm and its area is 60 cm². The area of the second triangle is 80 cm². Find the base of the second triangle (in cm).

Accepted Answer

Step 1: For the first triangle, Area = (1/2) × base × height → 60 = (1/2) × 12 × h → h = 10 cm. Step 2: For the second triangle with the same height h = 10 cm: 80 = (1/2) × base × 10 → base = 16 cm.

Question 14

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, so h = 12 cm.

Question 15

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. Find 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, so h = 12 cm.

SSC CHSL 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