The sides of a triangle are 16 cm, 12 cm, and 20 cm. Find :

(i) area of the triangle ;

(ii) height of the triangle, corresponding to the largest side ;

(iii) height of the triangle, corresponding to the smallest side.

#### Solution

Sides of ∆ are

a = 20 cm

b = 12 cm

c = 16 cm

`S = (a + b + c)/2`

= `(20 + 12 + 16)/2`

= `48/2 = 24`

area of Δ = `sqrt(s(s - a)(s - b)(s - c))`

= `sqrt(24(24 - 20)(24 - 12)(24 - 16))`

= `sqrt(24 xx 4 xx 12 xx 8)`

= `sqrt(12 xx 2 xx 4 xx 12 xx 2 xx 4)`

= `sqrt(12 xx 12 xx 4 xx 4 xx 2 xx 2)`

= `12 xx 4 xx 2 = 96 "cm"^2`

AD is height of Δ corresponding to largest side.

∴ `1/2 xx "BC" xx "AD" = 96`

`1/2 xx 20 xx "AD" = 96`

AD = `(96 xx 2)/20`

AD = 9.6 cm

BE is height of Δ corresponding to smallest side.

∴ `1/2 "AC" xx "BE" = 96`

`1/2 xx 12 xx "BE" = 96`

BE = `(96 xx 2)/12`

BE = 16 cm

(i) 96 cm^{2 }(ii) 9.6 cm (iii) 16 cm