Points \(A\) and \(B\) lie on sides \(PQ\) and \(PR\) of \(\triangle PQR\) such that \(PQ=12.5\,\text{cm}\), \(PA=5\,\text{cm}\), \(PB=4\,\text{cm}\) and \(BR=6\,\text{cm}\). Is \(AB\parallel QR\)? Give reasons.
Yes.
Step 1: Recall the Basic Proportionality Theorem (also called Thales' theorem). It says: If a line drawn through a triangle is parallel to one side, then it divides the other two sides in the same ratio.
Step 2: To check if \(AB \parallel QR\), we compare the ratios:
\[ \dfrac{PA}{PQ} \quad \text{and} \quad \dfrac{PB}{PR} \]
Step 3: First, calculate the length of \(PR\):
\(PR = PB + BR = 4 + 6 = 10\,\text{cm}\).
Step 4: Now find the ratios:
\[ \dfrac{PA}{PQ} = \dfrac{5}{12.5} \]
Divide numerator and denominator: \(5 \div 12.5 = 0.4\).
\[ \dfrac{PB}{PR} = \dfrac{4}{10} \]
Divide numerator and denominator: \(4 \div 10 = 0.4\).
Step 5: Both ratios are equal (0.4 = 0.4).
Step 6: Since the ratios are equal, by the Basic Proportionality Theorem, line \(AB\) is parallel to \(QR\).