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Question

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

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Solution

Perimeter = 144 cm and ratio of sides = 3 : 4 : 5
Sum of ratio terms = (3 + 4 + 5) = 12.
Let the lengths of the sides be a, b and c respectively.

Then, a=(144×312) cm=36 cm,b=(144×412) cm=48 cm and c=(144×412) cm=60 cm
s=12(a+b+c)=12(36+48+60) cm=72 cm.
By Heron's formula, the area of the triangle is given by
Δ=s(sa)(sb)(sc)
=72×36×24×12 cm2
=36×36×24×24 cm2
=(36×24) cm2=864 cm2
Hence, the area of the given triagnel is 864 cm2.
Let base = longest side = 60 cm and the correspoindig height =h cm
Then, area =(12×base×height) sq units
=(12×60×h) cm2=(30 h) cm2
30h=864 h=(86430)=28.8 cm
Hence, the height corresponding to the longest side is 28.8 cm

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