Similar Triangles: Finding Side PR Length

We are given two similar triangles, \(PQR\) and \(ABC\). The side lengths for triangle \(ABC\) are \(AB = 12 \text{ cm}\), \(AC = 15 \text{ cm}\), and \(CB = 21 \text{ cm}\). The side lengths for triangle \(PQR\) are \(PQ = 4 \text{ cm}\) and \(RQ = 7 \text{ cm}\). We need to find the length of side \(PR\).

Similarity Ratios

Since triangles \(PQR\) and \(ABC\) are similar (denoted as \(PQR \sim ABC\)), their corresponding sides are in proportion. The correspondence is \(P \leftrightarrow A\), \(Q \leftrightarrow B\), and \(R \leftrightarrow C\). Therefore, the ratios of corresponding sides are equal:

\(\frac{PQ}{AB} = \frac{QR}{BC} = \frac{PR}{AC}\)

Calculating the Ratio Factor

We can use the known sides to find the scale factor between the two triangles.

• Ratio using sides \(PQ\) and \(AB\):

  \(\frac{PQ}{AB} = \frac{4 \text{ cm}}{12 \text{ cm}} = \frac{1}{3}\)

• Ratio using sides \(RQ\) (which is the same as \(QR\)) and \(CB\) (which is the same as \(BC\)):

  \(\frac{RQ}{CB} = \frac{7 \text{ cm}}{21 \text{ cm}} = \frac{1}{3}\)

Both ratios confirm the scale factor is \(\frac{1}{3}\). Triangle \(PQR\) is smaller than triangle \(ABC\).

Finding the Length of PR

Now we use the similarity ratio involving the side \(PR\) and its corresponding side \(AC\):

\(\frac{PR}{AC} = \frac{1}{3}\)

Substitute the known length of \(AC = 15 \text{ cm}\):

\(\frac{PR}{15 \text{ cm}} = \frac{1}{3}\)

To find \(PR\), multiply both sides by \(15 \text{ cm}\):

\(PR = \frac{1}{3} \times 15 \text{ cm}\)

\(PR = 5 \text{ cm}\)

Thus, the length of side \(PR\) is \(5 \text{ cm}\).