Heron's formula, mensuration, areas

Question

A right triangle has perimeter 144 cm and hypotenuse 65 cm find its base and height. Also find its area using heron's formula

This is the question. If any one knows how to solve then please help

Answer 1

Given : Hypotenuse H=65 , Perimeter P=144 , therefore sum of Base B and Adjacent A is 144-65=79. Hence, we have following equations , A + B = 79 and A$^2$+ B$^2$ = 65$^2$. Substitute these in the equation (A + B)$^2$ = A$^2$ + B$^2$ + 2AB , ie , 79$^2$ = 65$^2$ + 2AB giving AB = 1008 . This implies A is 1008/B and substituting this in the eqn A+B=79 yields a quadratic equation, B$^2$ - 79B + 1008 = 0 . Solving this quadratic eqn gives B = 63 or 16. Say B=63 then A=16 . Now, substituting this in Heron's formula gives the Area = 504

Answer 2

Hint

The legs of the triangle, say $a,b,$ satisfy:

• $a+b=144-65$ (perimeter)
• $a^2+b^2=65^2$ (Pythagoras theorem)

Solve the system and you will get $a$ and $b.$

Then, since you know $a,b$ and the hypotenuse, just substutite in the Heron's formula.