Question

In \(\triangle ABC\) and \(\triangle PQR\), \(\angle A = \angle P\) and \(\angle B = \angle Q\). If AB = 4 cm, BC = 6 cm and PQ = 8 cm, then QR is

• A. 9 cm
• B. 10 cm
• C. 12 cm
• D. 3 cm

Hint:

Notice that the side \(PQ\) is exactly twice the corresponding side \(AB\) (\(8\text{ cm} = 2 \times 4\text{ cm}\)).
Because the triangles are similar, all sides of \(\triangle PQR\) must be exactly twice the corresponding sides of \(\triangle ABC\).
Thus, \(QR = 2 \times BC = 2 \times 6 = 12\text{ cm}\). This mental calculation takes only a few seconds.

Answer 1

The Correct Option is C

Step 1: Find the scale factor between the similar triangles.
Since $\angle A = \angle P$ and $\angle B = \angle Q$, by AA similarity $\triangle ABC \sim \triangle PQR$, so corresponding sides scale by the same factor. Using the known pair $AB$ and $PQ$: \[ \text{scale factor} = \frac{PQ}{AB} = \frac{8}{4} = 2 \]
Step 2: Apply this scale factor to the other known side.
Side $BC$ in the first triangle corresponds to side $QR$ in the second.
Step 3: Compute QR.
\[ QR = 2 \times BC = 2 \times 6 = 12 \text{ cm} \]
Step 4: State the result.
So $QR = 12$ cm, matching the corresponding sides proportion.
\[ \boxed{12 \text{ cm}} \]