Question

The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side.

Answer 1

Let the sides of a triangle are 3x, 4x and 5x.
Now, a = 3x, b = 4x and c = 5x
The perimeter 2s = 144
⇒ 3x + 4x + 5x = 144 [∵ a + b + c = 2s]
⇒ 12x = 144
⇒ x = 12
∴ sides of triangle are a = 3(x) = 36cm
b = 4(x) = 48 cm
c = 5(x) = 60 cm

Now semi perimeter s`1/2(a+b+c)=1/2(144)=72cm`

By heron’s formulas ∴ Area of Δle = `sqrt(s(s-a)(s-b)(s-c))`

`=sqrt(72(72-36)(72-48)(72-60)`

`=864cm^2`

Let l be the altitude corresponding to longest side,∴`1/2xx60xxl=864`

`⇒l=(864xx2)/60`

`⇒l=28.8cm`

Hence the altitude one corresponding long side = 28.8 cm