Question

in \\( \triangle a b c \\), which of the following is equal to \\( \frac{12}{13} \\)?
\\( \overline{p q} \\) is parallel to \\( \overline{x y} \\). what is the length of \\( \overline{r x} \\) in centimeters?

Explanation:

Step1: Use the property of similar triangles

Since \(\overline{PQ}\parallel\overline{XY}\), \(\triangle PQR\sim\triangle XYR\). For similar triangles, the ratios of corresponding sides are equal. That is \(\frac{PQ}{XY}=\frac{PR}{RX}\).
We know \(PQ = 4\mathrm{cm}\), \(XY=4\mathrm{cm}\), \(PR=(6 + RX)\mathrm{cm}\). Let \(RX=x\mathrm{cm}\), then \(PR=(6 + x)\mathrm{cm}\).

Step2: Set up the proportion

\(\frac{4}{4}=\frac{6 + x}{x}\). Simplifying, we get \(1=\frac{6 + x}{x}\), which implies \(x=6 + x\) (this is wrong). Let's use another property.
The correct proportion is \(\frac{PQ}{XY}=\frac{QR}{RY}=\frac{PR}{RX}\). We know \(PQ = 4\mathrm{cm}\), \(XY = 4\mathrm{cm}\), \(QR = 6\mathrm{cm}\), \(RY=4\mathrm{cm}\). Using \(\frac{QR}{RY}=\frac{PR}{RX}\), and \(PR=8 + RX\). Let \(RX=x\), then \(\frac{6}{4}=\frac{8 + x}{x}\).
Cross - multiply: \(6x=4(8 + x)\).

Step3: Solve the equation

Expand \(6x=4(8 + x)\) to \(6x=32+4x\).
Subtract \(4x\) from both sides: \(6x-4x=32+4x - 4x\), so \(2x=32\), \(x = 3\).

Answer:

\(3\)