Question

show that the length of the hypotenuse is \\(\sqrt{2}\\) times as long as each leg.
\\(a^2 + a^2 = c^2\\)
\\(\checkmark\\) 2. combine like terms.
\\(2a^2 = c^2\\)
\\(\checkmark\\) 3. take the principal square root of both sides.
\\(\sqrt{2a^2} = \sqrt{c^2}\\)

1. simplify \\(\boxed{c}\\) \\(\sqrt{2} = \boxed{2}\\) (with a dropdown menu for selection: 2, a, c) and a check button.

Explanation:

Step1: Start with Pythagorean theorem

For an isosceles right triangle, legs are equal: $a^2 + a^2 = c^2$

Step2: Combine like terms

$2a^2 = c^2$

Step3: Take principal square roots

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

Step4: Simplify square roots

$\sqrt{2}a = c$, or $c = a\sqrt{2}$

Answer:

The simplified equation is $\boldsymbol{c = a\sqrt{2}}$, so the first dropdown selects $\boldsymbol{a}$ and the second dropdown selects $\boldsymbol{a}$ (to write the relationship as $c = a\sqrt{2}$, matching that the hypotenuse $c$ is $\sqrt{2}$ times each leg $a$).