Question

if the angles of a triangle are $45^\circ$, $45^\circ$, and $90^\circ$,
show that the length of the hypotenuse is $\sqrt{2}$ times as long as each
leg.
$a^2 + a^2 = c^2$
✓ 2. combine like terms.
$2a^2 = c^2$
✓ 3. take the principal square root of both sides.
$\sqrt{2a^2} = \sqrt{c^2}$

1. simplify $\sqrt{2} = $

check

Explanation:

Step1: Apply Pythagorean theorem

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

Step2: Combine like terms

$2a^2 = c^2$

Step3: Take principal square roots

$\sqrt{2a^2} = \sqrt{c^2}$

Step4: Simplify both sides

$\sqrt{2}a = c$

Answer:

$c = a\sqrt{2}$, which shows the hypotenuse $c$ is $\sqrt{2}$ times as long as each leg $a$.