1. ## proof

complete the proof of the following:

consider a right triangle with hypotenuse c and legs a and b where a,b,and c are positive real numbers.

the right triangle is isosceles if and only if area=1/4(c^2)

i have not done an if and only proof before and would greatly appreciate it if someone could please help me to work through this

2. Originally Posted by BiGpO6790
complete the proof of the following:

consider a right triangle with hypotenuse c and legs a and b where a,b,and c are positive real numbers.

the right triangle is isosceles if and only if area=1/4(c^2)

i have not done an if and only proof before and would greatly appreciate it if someone could please help me to work through this
To prove an "iff" statement, you need to prove it both ways.

So, you need to show:

1. If the right angle triangle is isosceles then the area = $\frac{1}{4}c^2$

and

2. If the area $= \frac{1}{4}c^2$ then the right angle triangle is isosceles.

1. The area of the triangle is given by

$A = \frac{1}{2}BH$.

Since the triangle is isosceles, we have

$a = b$.

So $A = \frac{1}{2}a^2$.

By Pythagoras' Theorem, we have

$a^2 + b^2 = c^2$

$a^2 + a^2 = c^2$

$2a^2 = c^2$

$\frac{1}{2}a^2 = \frac{1}{4}c^2$.

So $A = \frac{1}{4}c^2$.

Try to prove Part 2...