Question

part a
what is the value of the ratio \\( \frac { y } { x } \\) ?

part b
what is the value of the ratio \\( \frac { h } { y } \\) ?

Explanation:

Step1: Find the horizontal side of the larger right - triangle

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 5\), \(c=13\), and \(b\) is the horizontal side.

$$b=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12$$

Step2: Use the property of similar triangles

Since \(\triangle PQR\) and the larger right - triangle (with sides \(5\), \(12\), \(13\)) are similar. For similar triangles, the ratios of corresponding sides are equal.
The ratio \(\frac{y}{x}\) is equal to the ratio of the vertical side to the horizontal side of the larger right - triangle. So \(\frac{y}{x}=\frac{5}{12}\)

Step3: For the ratio \(\frac{h}{y}\)

The ratio of the hypotenuse to the vertical side of the larger right - triangle is \(\frac{13}{5}\). Since \(\triangle PQR\) and the larger right - triangle are similar, \(\frac{h}{y}=\frac{13}{5}\)

Answer:

Part A: \(\frac{5}{12}\)
Part B: \(\frac{13}{5}\)