Question

z. simplify the answer, rationalize the denominator if necessary, our answer to decimal form. draw work here the square root symbol (√□), type
oot\

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the sides is \(1:1:\sqrt{2}\), where the hypotenuse \(c\) is related to the legs \(a\) (or \(b\)) by the formula \(c = a\sqrt{2}\). Here, \(c = 16\) and \(a=z\).

Step2: Solve for \(z\)

From \(c = a\sqrt{2}\), we can rewrite it as \(a=\frac{c}{\sqrt{2}}\). Substitute \(c = 16\) into the formula: \(z=\frac{16}{\sqrt{2}}\).
Rationalize the denominator: \(z=\frac{16\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{16\sqrt{2}}{2}=8\sqrt{2}\).
Convert to decimal: \(\sqrt{2}\approx1.414\), so \(z = 8\times1.414=11.312\).

Answer:

\(z = 8\sqrt{2}\approx11.31\)