Question

Find the area of a right angle triangle if the sum of its two sides containing the right angle is 14 cm and the hypotenuse is 10 cm, also verify the answer using Heron formula.

Answer 1

consider 2 sides of the right angled triangle as x and y
x+y=14-----1
x^2+y^2=10^2------2 (pythagores theorem)
cosider equation 1
y=14-x substitute this in equation 2
x^2+(14-x)^2=100
x^2+(14^2+2*14*x+x^2)=100
x^2+196+28x+x^2=100
2x^2+28x+96=0 on solving the equation
we get x=6 and 8
so take any of the value here im taking x=6 so y=14-6=8

area of a right angles triangle is = 1/2*b*h
=1/2*8*6
=24cm^2

using heron formula A= sqrt(s*(s-a)(s-b)(s-c)
s=(a+b+c)/2
=(6+8+10)/2=12
A= sqrt(12*(12-6)(12-8)(12-10))
=24cm^2

hence verified

hope it helps