Class IX Math NCERT Solution for Heron’s Formula 12.1

Exercise 12.1 (Page 202) .
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?
Ans. For an equilateral triangle with side ‘a’, area
       
       ∴ Each side of the triangle = a cm
       ∴ a + a + a = 180 cm
       ⇒ 3a = 180 cm
       
       Now, s = Semi–perimeter 
       ∴ Area of a triangle
           
       ∴ Area of the given triangle
       
       
       Thus, the area of the given triangle
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 (see Fig.). The advertisements yield an earning of Rs. 5000 per m2 per year. A company hired one of its walls for 3 months. How much rent did it pay?
Ans. The sides of the triangular wall are
       a = 122 m, b = 120 m, c = 22 m
       
       
3.   There 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 figure). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.
Ans. ∴ Sides of the wall are 15 m, 11 m and 6 m.
       ∴ a = 15 m, b = 11 m, c = 6 m
       
4.   Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter? is 42 cm.
Ans. Let the sides of the triangle be
       a = 18 cm, b = 10 cm and c = ?
       ∴ Perimeter (2s) = 42 cm
       
5.   Side of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm its area.
Ans. Perimeter of the triangle = 540 cm
       ⇒ Semi–perimeter of the triangle,
           
       ∴ The sides are in the ratio of 12 : 17 : 25.
       ∴ a = 12x cm, b = 17x cm, c = 25x cm
       ∴ 12x + 17x + 25x = 540
       ⇒ 54x = 540
       
       ∴ a = 12 × 10 = 120 cm
           b = 17 × 10 = 170 cm
           c = 25 × 10 = 250 cm
       ⇒ (s - a) = (270 - 120) cm = 150 cm
           (s - b) = (270 - 170) cm = 100 cm
           (s - c) = (270 - 250) cm = 20 cm
       ∴ Area of the triangle
       
6.   An isosceles triangle has perimeter 30 cm and each of the equal side is 12 cm. Find the area of the triangle.
Ans. Equal sides of the triangle are 12 cm each.
       Let the third side = x cm.
       Since, perimeter = 30 cm
       ∴ 12 cm + 12 cm + x cm = 30 cm
       ⇒ x = 30 - 12 - 12 cm
       ⇒ x = 6 cm
       Now, semi–perimeter 
       ∴ Area of the triangle
       

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