Question

In the below figure \(△PQR~△ABC\), then \(z+y=\) _____ .

• A. \(1+3\sqrt 3\)
• B. \(9+\sqrt 3\)
• C. \(7+3\sqrt 3\)
• D. \(4+3\sqrt 3\)

Answer 1

The Correct Option is D

To solve the problem, we are given that triangles \( \triangle PQR \sim \triangle ABC \). Since the triangles are similar, corresponding sides are proportional.

1. Understand the Corresponding Sides:
From the diagram, the corresponding sides are:
\( PQ \leftrightarrow AB, \quad QR \leftrightarrow BC, \quad PR \leftrightarrow AC \)

2. Use the Known Values from \( \triangle PQR \):
\( PQ = 3, \quad QR = y, \quad PR = 6 \)

From \( \triangle ABC \): \( AB = z, \quad BC = 4\sqrt{3}, \quad AC = 8 \)

3. Use the Ratio of Hypotenuses:
Since \( PR \) corresponds to \( AC \):
\[ \frac{PR}{AC} = \frac{6}{8} = \frac{3}{4} \]

4. Find Corresponding Sides:

\( PQ \leftrightarrow AB \Rightarrow \frac{PQ}{AB} = \frac{3}{z} = \frac{3}{4} \Rightarrow z = 4 \)

\( QR \leftrightarrow BC \Rightarrow \frac{QR}{BC} = \frac{y}{4\sqrt{3}} = \frac{3}{4} \Rightarrow y = 3\sqrt{3} \)

5. Add \( z + y \):
\[ z + y = 4 + 3\sqrt{3} \]

Final Answer:
The value of \( z + y \) is \( 4 + 3\sqrt{3} \).