Question

1. triangle with right angle, 45° angle, hypotenuse ( 8sqrt{2} ), sides labeled ( u ) and ( v )

Explanation:

Step1: Identify triangle type

This is a right - isosceles triangle (one right angle, one \(45^{\circ}\) angle, so the third angle is also \(45^{\circ}\), so the legs \(u\) and \(v\) are equal). For a right - isosceles triangle, if the hypotenuse is \(c\) and the legs are \(a\) (since \(a = b\) in a \(45 - 45-90\) triangle), the relationship is \(c=a\sqrt{2}\).

Step2: Solve for \(u\) and \(v\)

We know that the hypotenuse \(c = 8\sqrt{2}\). Using the formula \(c=a\sqrt{2}\), we can solve for \(a\) (where \(a = u=v\)).
From \(c=a\sqrt{2}\), we get \(a=\frac{c}{\sqrt{2}}\). Substitute \(c = 8\sqrt{2}\) into the formula:
\(a=\frac{8\sqrt{2}}{\sqrt{2}}\)
The \(\sqrt{2}\) in the numerator and denominator cancels out, so \(a = 8\). So \(u = 8\) and \(v = 8\).

Answer:

\(u = 8\), \(v = 8\)