Exercise 10.1 Solutions (Class 9)
Heron’s Formula
1. A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?
Part 1: Area in terms of ‘a’ using Heron’s Formula.
Sides are a, a, a. Perimeter = 3a.
Semi-perimeter (s) = 3a / 2.
Area = √(s(s-a)(s-b)(s-c))
= √((3a/2)(3a/2 - a)(3a/2 - a)(3a/2 - a))
= √((3a/2)(a/2)(a/2)(a/2)) = √(3a⁴/16) = (√3/4)a².
Part 2: Area when perimeter is 180 cm.
Perimeter = 3a = 180 cm => a = 60 cm.
Area = (√3/4)a² = (√3/4)(60)² = (√3/4)(3600) = 900√3 cm².
Answer: The area is 900√3 cm².
2. The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m. The advertisements yield an earning of ₹5000 per m² per year. A company hired one of its walls for 3 months. How much rent did it pay?
Step 1: Find the area of the triangular wall.
Sides a=122, b=22, c=120.
Semi-perimeter (s) = (122+22+120)/2 = 264/2 = 132 m.
Area = √(132(132-122)(132-22)(132-120))
= √(132 × 10 × 110 × 12) = √((12×11) × 10 × (11×10) × 12) = 12 × 11 × 10 = 1320 m².
Step 2: Calculate the rent.
Rent per year = Area × Rate = 1320 × 5000 = ₹6,600,000.
Rent for 3 months (1/4 of a year) = (1/4) × 6,600,000 = ₹1,650,000.
Answer: The company paid ₹1,650,000 in rent.
3. There is a slide in a park… If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.
Sides a=15, b=11, c=6.
Semi-perimeter (s) = (15+11+6)/2 = 32/2 = 16 m.
Area = √(16(16-15)(16-11)(16-6))
= √(16 × 1 × 5 × 10) = √800 = 20√2 m².
Answer: The area painted is 20√2 m².
4. Find the area of a triangle two sides of which are 18cm and 10cm and the perimeter is 42cm.
Step 1: Find the third side.
Perimeter = 18 + 10 + c = 42 => 28 + c = 42 => c = 14 cm.
Step 2: Find the area using Heron’s Formula.
Semi-perimeter (s) = Perimeter / 2 = 42 / 2 = 21 cm.
Area = √(21(21-18)(21-10)(21-14))
= √(21 × 3 × 11 × 7) = √((7×3) × 3 × 11 × 7) = 21√11 cm².
Answer: The area of the triangle is 21√11 cm².
5. Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540cm. Find its area.
Step 1: Find the lengths of the sides.
Let the sides be 12x, 17x, and 25x.
Perimeter = 12x + 17x + 25x = 54x = 540 cm => x = 10.
Sides are: a = 120 cm, b = 170 cm, c = 250 cm.
Step 2: Find the area using Heron’s Formula.
Semi-perimeter (s) = 540 / 2 = 270 cm.
Area = √(270(270-120)(270-170)(270-250))
= √(270 × 150 × 100 × 20) = √81000000 = 9000 cm².
Answer: The area of the triangle is 9000 cm².
6. An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.
Step 1: Find the third side.
Perimeter = 12 + 12 + c = 30 => 24 + c = 30 => c = 6 cm.
Step 2: Find the area using Heron’s Formula.
Sides are a=12, b=12, c=6.
Semi-perimeter (s) = 30 / 2 = 15 cm.
Area = √(15(15-12)(15-12)(15-6))
= √(15 × 3 × 3 × 9) = √1215 = 9√15 cm².
Answer: The area of the triangle is 9√15 cm².
