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 $\boxed{\\ \\ } \sqrt{2} = \boxed{\\ \\ }$

Explanation:

Step1: Apply Pythagorean theorem

For a right triangle with legs $a, a$ and hypotenuse $c$,
$$a^2 + a^2 = c^2$$

Step2: Combine like terms

Add the terms on the left side:
$$2a^2 = c^2$$

Step3: Take principal square roots

Apply square root to both sides:
$$\sqrt{2a^2} = \sqrt{c^2}$$

Step4: Simplify the square roots

Simplify each side using $\sqrt{x^2}=x$ for positive $x$:
$$a\sqrt{2} = c$$
Rearrange to isolate the ratio:
$$\frac{c}{a} = \sqrt{2}$$

Answer:

$\frac{c}{a} = \sqrt{2}$ (or $c = a\sqrt{2}$, showing the hypotenuse $c$ is $\sqrt{2}$ times each leg $a$)