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Answer :

Suppose that the sides in cm, are 12x, 17x and 25x.

Then, we know that

12x + 17x + 25x = 540

(Given, Perimeter of the triangle)

54x = 540

\(\Rightarrow \) x = 10

So, the sides of the triangle are 12 × 10cm, 17 × 10cm, 25 × 10cm \(\Rightarrow \) 120cm, 170cm and 250cm.

Now, we know that,

\(s = \frac{a + b + c}{2}\)

\(\therefore \) \(s = \frac{120 + 170 + 250}{2} = \frac{540}{2} = 270cm\)

Now, Area of triangle

= \(\sqrt{270(270 - 120)(270 - 170)(270 - 250)}\)

(Since, Heron's formula [area = \(\sqrt{s(s - a)(s - b)(s - c)}\)])

= \(\sqrt{270 × 150 × 100 × 20}\)

= \(\sqrt{27 × 10 × 3 × 15 × 10 × 100 × 20}\)

= \(100\sqrt{27 × 15 × 10 × 2} {cm}^2\)

= \(100\sqrt{9 × 3 × 3 × 5 × 10 × 2} {cm}^2\)

= \(100\sqrt{9 × 3 × 3 × 10 × 10} {cm}^2\)

= 100 × 10 × 9

Therefore, area of the given triangle is 9000\({cm}^2\)

- 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?
- The triangular side walls of a flyover have been used for advertisements. The sides of the wails are 122 m, 22 m and 120 m (see figure). The advertisements yield earnings of Rs. 5000 per \({m}^2\) per year. A company hired one of its walls for 3 months. How much rent did it pay?
- There is a slide in a park. One of its side walls has been painted in same 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.
- Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
- An isosceles triangle has perimeter 30 cm arid each of the equal sides is 12 cm. Find the area of the triangle.

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