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: Understanding the given problem:
We are given a triangle \( \triangle PQR \) with a line \( l \) intersecting the sides \( PQ \) and \( PR \) at points \( L \) and \( M \), respectively. It is also given that \( LM \parallel QR \). We need to find the length of \( PM \).
The given information is:
- \( PL = 5.7 \, \text{cm} \)
- \( PQ = 15.2 \, \text{cm} \)
- \( MR = 5.5 \, \text{cm} \)

Step 2: Using the Basic Proportionality Theorem (Thales' Theorem):
Since \( LM \parallel QR \), by the Basic Proportionality Theorem (also known as Thales' Theorem), we know that: \[ \frac{PL}{PQ} = \frac{PM}{PR} \] We are given that \( PL = 5.7 \, \text{cm} \) and \( PQ = 15.2 \, \text{cm} \), so: \[ \frac{PL}{PQ} = \frac{5.7}{15.2} \] This gives the proportion between the corresponding segments on \( PQ \) and \( PR \). Now, we need to find \( PM \) using the given segment \( MR \). Since the total length of \( PR \) is the sum of \( PM \) and \( MR \), we can express the total length of \( PR \) as: \[ PR = PM + MR = PM + 5.5 \, \text{cm} \] Substitute the known ratio: \[ \frac{5.7}{15.2} = \frac{PM}{PM + 5.5} \] Now, solve for \( PM \):

Step 3: Solving the equation:
Cross-multiply to get rid of the fractions: \[ 5.7 (PM + 5.5) = 15.2 \times PM \] Simplifying: \[ 5.7 \, PM + 5.7 \times 5.5 = 15.2 \, PM \] \[ 5.7 \, PM + 31.35 = 15.2 \, PM \] Now, subtract \( 5.7 \, PM \) from both sides: \[ 31.35 = 15.2 \, PM - 5.7 \, PM \] \[ 31.35 = 9.5 \, PM \] Solve for \( PM \): \[ PM = \frac{31.35}{9.5} = 3.3 \, \text{cm} \]

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