Question

solve the right triangle.
write your answers in simplified, rationalized form. do not round.
uv =
m∠v =
m∠u =

Explanation:

Step1: Calculate hypotenuse $UV$

Use Pythagorean theorem:
$$UV = \sqrt{(\sqrt{31})^2 + (\sqrt{31})^2} = \sqrt{31 + 31} = \sqrt{62}$$

Step2: Find $m\angle V$

Use tangent ratio:
$$\tan(V) = \frac{TU}{TV} = \frac{\sqrt{31}}{\sqrt{31}} = 1$$
$$m\angle V = \tan^{-1}(1) = 45^\circ$$

Step3: Find $m\angle U$

Sum of triangle angles is $180^\circ$:
$$m\angle U = 180^\circ - 90^\circ - 45^\circ = 45^\circ$$

Answer:

$UV = \sqrt{62}$
$m\angle V = 45^\circ$
$m\angle U = 45^\circ$