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

Options

A. `26.8` cm
B. `24.8` cm
C. `18.8` cm
D. `28.8` cm

Answer

D

Explanation

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=(144xx(3)/(12))cm = 36 cm, b=(144xx(4)/(12))cm=48 cm`
and `c =(144xx(5)/(12))cm = 60 cm`.
`therefore" "s=(1)/(2)(a+b+c)=(1)/(2)(36+48+60)cm=72 cm`.
`therefore" "(s-a)=(72-36)cm=36 cm ,(s-b)=(72-48)cm-24cm and (s-c)=(72-60)cm = 12 cm`.
(i) By Heron's formula, the area of the triangle is given by
`Delta=sqrt(s(s-a)(s-b)(s-c))`
`=sqrt(72 xx 36 xx 24 xx 12)cm^(2)=sqrt(36 xx 24 xx 24)cm^(2)`
`=(36 xx 24)cm^(2)=864 cm^(2)`. Hence, the area of the given triangle is 864 `cm^(2)`.
(ii) Let base = longest side = 60 cm and the corresponding height = h cm.
Then, area `=((1)/(2) xx "base" xx "height")` sq units
`=((1)/(2) xx 60 xx h)cm^(2) = (30h)cm^(2)`.
`therefore" "30h = 864 rArr h = ((864)/(30)) rArr "height" = 28.8 cm`.
Hence, the required height is 28.8 cm.