Question

A line l intersects the sides PQ and PR of a \(\triangle\) PQR at L and M respectively such that LM || QR. If PL = 5·7 cm, PQ = 15·2 cm and MR = 5·5 cm, then the length of PM (in cm) is :

• A. 3
• B. 1.8
• C. 2.5
• D. 3.3

Answer 1

The Correct Option is D

Step 1: Understand the given problem:
We are given a triangle \( \triangle PQR \), and a line \( l \) intersects the sides \( PQ \) and \( PR \) at points \( L \) and \( M \), respectively, such that \( LM \parallel QR \). We are given the following measurements: - \( PL = 5.7 \, \text{cm} \) - \( PQ = 15.2 \, \text{cm} \) - \( MR = 5.5 \, \text{cm} \) We are tasked with finding the length of \( PM \).

Step 2: Apply the Basic Proportionality Theorem (Thales' Theorem):
Since \( LM \parallel QR \), we can apply the Basic Proportionality Theorem (also known as Thales' Theorem), which states that if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. Therefore, we have the following proportion: \[ \frac{PL}{PQ} = \frac{PM}{PR} \] We are given that \( PL = 5.7 \, \text{cm} \) and \( PQ = 15.2 \, \text{cm} \). Also, we know that the total length of \( PR \) is \( PM + MR \). From the given information, \( MR = 5.5 \, \text{cm} \), so we can express \( PR \) as: \[ PR = PM + 5.5 \, \text{cm} \] Now, substitute the known values into the proportion: \[ \frac{5.7}{15.2} = \frac{PM}{PM + 5.5} \]

Step 3: Solve the proportion:
Cross-multiply to solve for \( PM \): \[ 5.7 \times (PM + 5.5) = 15.2 \times PM \] Simplify the equation: \[ 5.7 \times PM + 5.7 \times 5.5 = 15.2 \times PM \] \[ 5.7 \times PM + 31.35 = 15.2 \times PM \] Now, move all terms involving \( PM \) to one side: \[ 31.35 = 15.2 \times PM - 5.7 \times PM \] \[ 31.35 = 9.5 \times PM \] Solve for \( PM \): \[ PM = \frac{31.35}{9.5} = 3.3 \, \text{cm} \]

Step 4: Conclusion:
The length of \( PM \) is \( \boxed{3.3 \, \text{cm}} \).