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CBSE Class 9 Mathematics Question 5 of 6

Heron's Formula — Question 5

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Question

Question 5

Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.

Answer

Given,

Sides of a triangle are in the ratio of 12 : 17 : 25.

Let sides of the triangle are :

a = 12x, b = 17x and c = 25x.

Given,

Perimeter = 540 cm

∴ 12x + 17x + 25x = 540 cm

⇒ 54x = 540 cm

⇒ x = 54054\dfrac{540}{54}

⇒ x = 10 cm

⇒ a = 12 × 10 = 120 cm,

⇒ b = 17 × 10 = 170 cm,

⇒ c = 25 × 10 = 250 cm.

Semi perimeter (s) = Perimeter of triangle2=5402\dfrac{\text{Perimeter of triangle}}{2} = \dfrac{540}{2} = 270 cm.

By Heron's formula,

Area of triangle (A) = s(s−a)(s−b)(s−c)\sqrt{s(s - a)(s - b)(s - c)} sq.units

Substituting values we get :

A=270(270−120)(270−170)(270−250)=270×150×100×20=81000000=9000 cm2.A = \sqrt{270(270 - 120)(270 - 170)(270 - 250)} \\[1em] = \sqrt{270 \times 150 \times 100 \times 20} \\[1em] = \sqrt{81000000} \\[1em] = 9000 \text{ cm}^2.

Hence, area of triangle = 9000 cm2.

Heron's Formula - Interactive Study Notes | Bright Tutorials
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Mathematics | Heron's FormulaWeb Content • Interactive Notes

Heron’s Formula — Interactive Study Guide

The Formula

s = (a+b+c)/2
Area = √[s(s−a)(s−b)(s−c)]

Works for ANY triangle. No need to know the height!

Step-by-Step Calculation

Sides: 5, 12, 13

s = (5+12+13)/2 = 15

s−5 = 10, s−12 = 3, s−13 = 2

Area = √(15×10×3×2) = √900 = 30 sq units

Verify: This is a right triangle (5²+12²=13²), so area = ½×5×12 = 30. Matches!

Quick Self-Check

  1. Find area of equilateral Δ with side 6. (s=9, A=√(9×3×3×3)=9√3)
  2. Find area of triangle with sides 3, 4, 5. (6 sq units)

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