Question 1

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: Area = (1/2) × 12 × 8 = (1/2) × 96 = 48 cm². Therefore, the answer is 48 cm².

Question 2

An equilateral triangle has a side length of 6 cm. What is its perimeter?

Accepted Answer

Step 1: An equilateral triangle has all three sides equal. Step 2: Perimeter = 3 × side = 3 × 6 = 18 cm. Therefore, the answer is 18 cm.

Question 3

A right-angled triangle has legs of 5 cm and 12 cm. What is the length of its hypotenuse?

Accepted Answer

Step 1: Use the Pythagorean theorem: c² = a² + b². Step 2: c² = 5² + 12² = 25 + 144 = 169. Step 3: c = √169 = 13 cm. Therefore, the answer is 13 cm.

Question 4

A triangle has an area of 36 cm² and a base of 9 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 × 36) / 9 = 72 / 9 = 8 cm. Therefore, the answer is 8 cm.

Question 5

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 6

An equilateral triangle has a side length of 8 cm. What is the ratio of its area to the area of an equilateral triangle with side length 4 cm?

Accepted Answer

Step 1: For an equilateral triangle with side a, Area = (√3 / 4) × a². Step 2: Area of first triangle = (√3 / 4) × 8² = (√3 / 4) × 64 = 16√3 cm². Step 3: Area of second triangle = (√3 / 4) × 4² = (√3 / 4) × 16 = 4√3 cm². Step 4: Ratio = 16√3 / 4√3 = 4 / 1 = 4:1. Therefore, the ratio is 4:1.

Question 7

A triangle has vertices at coordinates A(0, 0), B(8, 0), and C(4, 6). Using the coordinate formula, find the area of triangle ABC.

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 8

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.

Question 9

Two similar triangles have corresponding sides in the ratio 3:5. If the area of the smaller triangle is 36 cm², what is the area of the larger triangle (in cm²)?

Accepted Answer

Step 1: When two triangles are similar, the ratio of their areas equals the square of the ratio of their corresponding sides. Step 2: If the side ratio is 3:5, then the area ratio is (3)²:(5)² = 9:25. Step 3: Let the area of the larger triangle be A. Then 36/A = 9/25. Step 4: A = (36 × 25)/9 = 900/9 = 100 cm².

Question 10

A triangle with sides 20 cm, 21 cm, and 29 cm is inscribed in a circle. What is the radius of the circumscribed circle (in cm)?

Accepted Answer

Step 1: Use Heron's formula to find the area. Semi-perimeter s = (20 + 21 + 29)/2 = 35 cm. Step 2: Area = √[35(35-20)(35-21)(35-29)] = √[35 × 15 × 14 × 6] = √44100 = 210 cm². Step 3: The circumradius formula is R = (abc)/(4A), where a, b, c are the sides and A is the area. Step 4: R = (20 × 21 × 29)/(4 × 210) = 12180/840 = 14.5 cm.

Question 11

In triangle ABC, the median from vertex A to side BC has length 9 cm. If the median from vertex B to side AC has length 12 cm, and the median from vertex C to side AB has length 15 cm, what is the area of the triangle (in cm²)?

Accepted Answer

Step 1: Use the median-based area formula: Area = (4/3)√[s_m(s_m - m_a)(s_m - m_b)(s_m - m_c)], where s_m is the semi-sum of the medians. Step 2: s_m = (9 + 12 + 15)/2 = 18 cm. Step 3: Area = (4/3)√[18(18-9)(18-12)(18-15)] = (4/3)√[18 × 9 × 6 × 3] = (4/3)√2916. Step 4: √2916 = 54 cm. Step 5: Area = (4/3) × 54 = 72 cm².

SSC GD Constable 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