Calculate the area of the triangle with the following data

Question

How do I calculate the area of the triangle when I know that the length of its hypotenuse is 8 and the sum of the legs is 10?

Answer 1

$a^2 + b^2 = 8^2$

$a + b = 10$

So $(a+b)^2 = 10^2$

So $a^2 + 2ab + b^2 = 10^2$.

So $(a^2 + 2ab + b^2) -(a^2 + b^2) = 10^2 - 8^2$

$2ab = 10^2 - 8^2$

$2ab = (10-8)(10+8)$

$ab = \frac{(10-8)(10+8)}2$.

So $Area = \frac 12 ab = \frac{(10-8)(10+8)}{2*2}$

Answer 2

You have that $$ c=8\qquad a+b=10 $$ and the area can be calculated as $$ A=\frac{ab}{2}. $$ Now $$ (a+b)^2=100=a^2+2ab+b^2\Rightarrow ab=-\frac{a^2+b^2}{2} $$ Can you figure out the rest? :)

Answer 3

WLOG the legs can be $8\cos t,8\sin t$

$10=8(\cos t+\sin t)$

Square both sides

Now, the area $$=\dfrac{8^2\sin t\cos t}2$$