Question

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

Explanation:

Step1: Determine angle T

In a right triangle, the sum of angles is \(180^\circ\). Given \(\angle V = 90^\circ\) and \(\angle U = 45^\circ\), so \(m\angle T=180^\circ - 90^\circ - 45^\circ = 45^\circ\).

Step2: Identify triangle type

Since \(\angle U=\angle T = 45^\circ\), \(\triangle UVT\) is an isosceles right triangle, so \(UV = TV\).

Step3: Use trigonometry for sides

Let \(UV = TV = x\). The hypotenuse \(UT = 8\sqrt{2}\). Using the Pythagorean theorem \(UV^{2}+TV^{2}=UT^{2}\), substitute \(UV = TV = x\):
\(x^{2}+x^{2}=(8\sqrt{2})^{2}\)
\(2x^{2}=64\times2\)
\(x^{2}=64\)
\(x = 8\) (since length is positive).

Answer:

\(TV = 8\)
\(UV = 8\)
\(m\angle T = 45\)