Chapter 10: Heron's Formula Exercise & In-Text Questions & Answers
1A 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?
Answer & Explanation:
Given: Equilateral triangle ⇒ all sides = a.
Perimeter = 3a = 180 cm ⇒ a = 60 cm.
Semi-perimeter s = 180/2 = 90 cm.
Using Heron’s formula:
Area = √[s(s−a)(s−a)(s−a)] = √[90×30×30×30] = √[2430000] = 900√3 cm². Reference: Based on example of equilateral triangle with side 10 cm in Section 10.1.
2The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see Fig. 10.6). 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?
Answer & Explanation:
Sides: a = 122 m, b = 22 m, c = 120 m.
s = (122 + 22 + 120)/2 = 132 m.
Area = √[132(132−122)(132−22)(132−120)] = √[132×10×110×12] = √[1742400] = 1320 m².
Annual earning = 1320 × ₹5000 = ₹66,00,000.
For 3 months = ₹66,00,000 × (3/12) = ₹16,50,000. Reference: Exercise 10.1, Question 2.
3There is a slide in a park. One of its side walls has been painted in some colour with a message "KEEP THE PARK GREEN AND CLEAN" (see Fig. 10.7). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.
Answer & Explanation:
Sides: a = 15 m, b = 11 m, c = 6 m.
s = (15 + 11 + 6)/2 = 16 m.
Area = √[16(16−15)(16−11)(16−6)] = √[16×1×5×10] = √800 = 20√2 m² ≈ 28.28 m². Reference: Exercise 10.1, Question 3.
4Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Answer & Explanation:
Given: a = 18 cm, b = 10 cm, perimeter = 42 cm.
Third side c = 42 − (18 + 10) = 14 cm.
s = 42/2 = 21 cm.
Area = √[21(21−18)(21−10)(21−14)] = √[21×3×11×7] = √4851 = 21√11 cm². Reference: Similar to Example 1 in Section 10.1.
5Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.
Answer & Explanation:
Let sides be 12x, 17x, 25x.
Perimeter = 12x + 17x + 25x = 54x = 540 ⇒ x = 10.
Sides = 120 cm, 170 cm, 250 cm.
s = 540/2 = 270 cm.
Area = √[270(270−120)(270−170)(270−250)] = √[270×150×100×20] = √81000000 = 9000 cm². Reference: Similar to Example 3 in Section 10.1.
6An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.
Answer & Explanation:
Equal sides = 12 cm each ⇒ third side = 30 − 24 = 6 cm.
s = 30/2 = 15 cm.
Area = √[15(15−12)(15−12)(15−6)] = √[15×3×3×9] = √1215 = 9√15 cm². Reference: Similar to isosceles triangle example in Section 10.1 (sides 5,5,8).