Question

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

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

Hint:

Always pay close attention to the order of vertices in a similarity statement!
A common mistake is writing $\frac{AB}{PQ}$ or $\frac{AB}{QR}$ arbitrarily.
Write down the letters of the triangles vertically aligned to match them up:
A $\rightarrow$ Q
B $\rightarrow$ R
C $\rightarrow$ P
This ensures you write the ratios correctly: $AB$ corresponds to $QR$, and $BC$ corresponds to $RP$.

Answer 1

The Correct Option is D

Step 1: Find the scale factor between the two triangles using the pair of sides we already know fully.
Since $\Delta ABC \sim \Delta QRP$, side $BC$ corresponds to side $RP$. We are given $BC = 5$ cm and $PR = 2$ cm, so the scale factor $k$ from $\Delta QRP$ to $\Delta ABC$ is:
\[ k = \frac{BC}{RP} = \frac{5}{2} = 2.5 \]

Step 2: Use this single scale factor on the other pair of corresponding sides, instead of writing a fresh proportion.
Since $AB$ corresponds to $QR$, and $\Delta ABC$ is the "bigger" triangle scaled up by $k$ from $\Delta QRP$:
\[ AB = k \times QR \]

Step 3: Substitute the known value of AB and solve for QR.
\[ 9 = 2.5 \times QR \]
\[ QR = \frac{9}{2.5} \]

Step 4: Simplify the division.
\[ QR = \frac{9}{2.5} = \frac{90}{25} = \frac{18}{5} = 3.6 \text{ cm} \]

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