The perimeter of an isosceles triangle is `32` cm. The ratio of the equal side to its base is `3:2`. Find the area of the triangle.

Options

A. `31sqrt2cm^(2)`
B. `32sqrt2cm^(2)`
C. `30sqrt2cm^(2)`
D. `42sqrt2cm^(2)`

Answer

B

Explanation

Let `ABC` be an isoceles triangle with perimeter `32 cm.` We have, ratio of equal side to its base is `13:2.`
`"Let sides of triangle be "AB=AC=3x,BC=2x`
`because"Perimeter of a triangle"=32 m`
`"Now", " "3x+3x+2x=32`
`rArr" "8x+32`
`rArr" "x+4`
`therefore" "AB=AC=3xx4=12cm`
`"and"" "BC=2x=2xx4=8cm`
The sides of a triangle are a =12 cm, b=12 cm
`"and"" "C=8cm`.
`therefore`Semi-perimeter of an isosceles triangle,
`s=(a+b+c)/(2)`
`=(12+12+8)/(2)=(32)/(2)=16cm`
`therefore"Area of an isosceles" triangleABC=sqrt(s(s-a)(s-b)(s-c))" "["by Heron's formula"]`
`=sqrt(16(16-12)(16-12)(16-8))=sqrt(16xx4xx4xx8)`
`rArr " "=4xx4xx2sqrt2cm^(2)=32sqrt2cm^(2)`
Hence, the area of an isosceles triangle is `32sqrt2cm^(2)`.