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.

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#### Solution

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

Concept: Area of a Triangle by Heron's Formula

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