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.

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Solution

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+214x+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 = 1/2bh
=1/286
=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

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