Question

fill in the blank 1 point
if ( x = 9sqrt{2} ), find ( z ) and ( y ).
( z = ) choose your answer... ( y = ) choose your answer...

Explanation:

Step1: Identify the triangle type

This is a right - isosceles triangle (since one angle is \(90^{\circ}\) and another is \(45^{\circ}\), so the third angle is also \(45^{\circ}\)). In a right - isosceles triangle, the two legs (the sides opposite the \(45^{\circ}\) angles) are equal. So, \(x = z\) or \(x\) and \(z\) are the legs.
Given \(x = 9\sqrt{2}\), since it's a right - isosceles triangle, \(z=x = 9\sqrt{2}\) (because the two non - hypotenuse sides are equal in a \(45 - 45-90\) triangle).

Step2: Find the hypotenuse \(y\)

For a \(45 - 45-90\) triangle, the hypotenuse \(y\) is related to the leg length \(a\) (where \(a=x = z\)) by the formula \(y=a\sqrt{2}\).
Substitute \(a = 9\sqrt{2}\) into the formula: \(y=9\sqrt{2}\times\sqrt{2}\).
We know that \(\sqrt{2}\times\sqrt{2}=2\), so \(y = 9\times2=18\).

Answer:

\(z = 9\sqrt{2}\), \(y = 18\)