Question

It is given that \(\Delta ABC \sim \Delta QRP\) such that \(AB = 9\text{ cm}\), \(BC = 5\text{ cm}\) and \(PR = 2\text{ cm}\). Length of side \(QR\) is :

• A. \(0.9\text{ cm}\)
• B. \(\frac{5}{18}\text{ cm}\)
• C. \(\frac{10}{9}\text{ cm}\)
• D. \(3.6\text{ cm}\)

Hint:

Pay close attention to the order of letters in the similarity statement \(\Delta ABC \sim \Delta QRP\).
The order of the letters directly dictates which sides correspond to one another.
Do not assume standard matching like \(AB\) with \(PQ\) unless stated by the exact vertex correspondence.

Answer 1

The Correct Option is D

Step 1: Find the scale factor of the similarity first.
Since $\Delta ABC \sim \Delta QRP$, every side of $\Delta ABC$ is the same multiple of the matching side of $\Delta QRP$. Call this scale factor $k$, so $\dfrac{AB}{QR} = \dfrac{BC}{RP} = k$.

Step 2: Compute $k$ from the side we already know both values of.
We know $BC = 5\text{ cm}$ and its match $RP = 2\text{ cm}$ (recall $PR$ and $RP$ name the same segment), so:
\[ k = \frac{BC}{RP} = \frac{5}{2} \]

Step 3: Use the scale factor to get $QR$.
Since $AB = k \cdot QR$, we get $QR$ by dividing instead of cross-multiplying:
\[ QR = \frac{AB}{k} = \frac{9}{\frac{5}{2}} = 9 \times \frac{2}{5} = \frac{18}{5} \]
\[ QR = 3.6\text{ cm} \]

Final Answer:
The length of side $QR$ is 3.6 cm, matching option (D).
\[ \boxed{QR = 3.6\text{ cm}} \]