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: Understanding the Question:
This question is from Geometry, specifically dealing with Similar Triangles.
We are given information about equal angles in two triangles and some side lengths. We need to calculate a missing side length of the second triangle.

Step 2: Key Formula or Approach:
According to the Angle-Angle (AA) Similarity Criterion, if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
If \(\triangle ABC \sim \triangle PQR\), then the ratios of their corresponding sides are equal:
\[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} \]

Step 3: Detailed Explanation:
In \(\triangle ABC\) and \(\triangle PQR\):
\[ \angle A = \angle P \]
\[ \angle B = \angle Q \]
By the AA similarity criterion:
\[ \triangle ABC \sim \triangle PQR \]
Since the triangles are similar, their corresponding sides are proportional. Thus, we can write:
\[ \frac{AB}{PQ} = \frac{BC}{QR} \]
We are given:
\[ AB = 4\text{ cm} \]
\[ BC = 6\text{ cm} \]
\[ PQ = 8\text{ cm} \]
Substitute these values into the proportion:
\[ \frac{4}{8} = \frac{6}{QR} \] Simplify the left fraction:
\[ \frac{1}{2} = \frac{6}{QR} \] Cross-multiply to solve for QR:
\[ QR = 6 \times 2 \]
\[ QR = 12\text{ cm} \]

Step 4: Final Answer:
The length of the side QR is \(12\text{ cm}\).