Question

I need help on geometry proofs

Answer 1

The hypotenuse has got to have the longest side in a right angled triangle.

Say, A=Adjacent, O=Opposite, H=Hypotenuse

A²+O²=H²

Therefore,

A²<H²

O²<H²

As this is the case:

H>A

H>O

Answer 2

Other proof:

Look at the picture.

$sin\alpha\in (0;\ 1)\\\\sin\alpha=\frac{b}{h} < 1\ therefore\ h > b\\\\cos\alpha\in(0;\ 1)\\\\cos\alpha=\frac{a}{h} < 1\ therefore\ h > a\\\\the\ conclusion:\ h > a\ and\ h > b\ \Rightarrow h\ has\ to\ be\ the\ longest\ side\\of\ a\ right\ triangle.$