Question

find the area of the triangle whose sides are 42cm,34cm and 20cm. Also find the height corresponding to the longest side.

Answer 1

Semiperimeter (s) = (42 + 34 + 20 ) / 2
So it is = 96 / 2 = 48 cm.

Using herons formula = (√s )(√s-a )(√s-b) (√s-c)
By  substituting s and a , b , c   i.e  the given sides.
We have √48 × √6 × √14 × √28
So √ (4 × 6 × 2) × √6 × √(7 × 2) × √(7 × 4)
Simplifying the numbers we have area = 4 × 6 × 2 × 7 = 336 cm ²
Hence area is 336 cm²

Now height corresponding to longest side implies that base is 42 cm and area remains same 

So area of triangle is  1/ 2 b × h
336 = 1/ 2 ×  42 × H
HENCE H = 16 cm

So height is 16 cm .

Answer 2

make a trianlge ABC of 1 side=34cm one side=20 cm n the last side(base of the triangle) of 42cm

now use the heron's formula..according to the herons formula area of a triangle=under root [s(s-1st side of trianlge)(s-2nd side of triangle)(s-3rd side of trianle)].here s denotes the half of perimeter...perimeter is equals to the sum of all the sides of the triangle...that is 34+20+42 and it equals to 96..!!
 S is half of the perimeter........here our perimeter is 96.........so its half is 48...!!!
now just put your values in the formula
under root[48(48-34)(48-20)(48-42)]
under root[48*6*14*28]
under root(112896)
=336 cm.
it is given in the question to find the height corresponding to the largest side...here the largest side is 42cm....so it means that we have to take 42cm as our base......now area of triangle is also 1/2(half)*base of the triangle*height of the triangle

A=1/2*B*H
336=1/2*42*H
336=21*H
H=336/21
H=16cm..!!