Question

part b
how can you find the length of \\( \overline { r t } \\) using similarity? explain your reasoning.
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question 3
what is wx? explain your reasoning.

Explanation:

Step1: Analyze triangle \( \triangle YZX \)

In \( \triangle YZX \), since \( \angle Z = 90^{\circ}\), \( \angle X = 45^{\circ}\), then \( \angle ZYX=45^{\circ}\). By the property of isosceles right - triangle (\( \text{If in a right - triangle, two angles are equal to }45^{\circ}\text{, then the legs are equal}\)), we have \( ZX = ZY\). Given \( ZY = 10\), so \( ZX=10\).

Step2: Analyze triangle \( \triangle WZY \)

In \( \triangle WZY \), \( \angle WZY = 90^{\circ}\), \( \angle W = 30^{\circ}\), \( \angle WYZ=60^{\circ}\). We know that in a \(30 - 60-90\) triangle, if the side opposite the \(30^{\circ}\) angle is \(a\), the side opposite the \(60^{\circ}\) angle is \(a\sqrt{3}\), and the side opposite the \(90^{\circ}\) angle is \(2a\). Here, the side opposite the \(30^{\circ}\) angle is \(ZY = 10\), then \(WZ = 10\sqrt{3}\).

Step3: Calculate \( WX \)

Since \( WX=WZ + ZX\), substituting \(WZ = 10\sqrt{3}\) and \(ZX = 10\), we get \(WX=10\sqrt{3}+ 10=10(\sqrt{3}+1)\approx10(1.732 + 1)=27.32\).

Answer:

\(WX = 10(\sqrt{3}+1)\approx27.32\)