Question

proving similar triangles practice
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if △ayb ~ △xyz, ab = 5, ay = 12, and ay = ax, what is the length of yz?
(1 point)
yz=
incorrect

• since △ayb ~ △xyz, the lengths of sides yb and yz are proportional. calculate yb and use the fact that ay = ax to find yz

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle AYB\sim\triangle XYZ\), the ratios of corresponding sides are equal. That is \(\frac{AY}{XY}=\frac{AB}{XZ}\). Given \(AY = AX = 12\), then \(XY=AY + AX=12 + 12=24\).

Step2: Set up the proportion

We know \(\frac{AY}{XY}=\frac{AB}{XZ}\), substituting the values we have \(\frac{12}{24}=\frac{5}{XZ}\). Cross - multiply gives \(12\times XZ=24\times5\), so \(XZ = 10\).

Step3: Use Pythagorean theorem (for right - angled similar triangles, ratio of sides)

Another way, since \(\triangle AYB\sim\triangle XYZ\), the ratio of similarity \(k=\frac{AY}{XY}=\frac{12}{24}=\frac{1}{2}\). Let \(YZ\) be the hypotenuse of \(\triangle XYZ\) and \(YB\) be the hypotenuse of \(\triangle AYB\).
We can also use the ratio of sides. In right - angled triangles \(\triangle AYB\) and \(\triangle XYZ\) (right - angled at \(A\) and \(X\) respectively), \(\frac{AY}{XY}=\frac{AB}{XZ}=\frac{YB}{YZ}\).
Since \(\frac{AY}{XY}=\frac{1}{2}\), and in right - angled triangle \(\triangle AYB\), if \(AY = 12\), \(AB = 5\), then by Pythagorean theorem \(YB=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\).
Since \(\frac{YB}{YZ}=\frac{1}{2}\), then \(YZ = 2\times YB\).

Answer:

\(26\)