$A$ and $B$ are points on the sides $PQ$ and $PR$ of a triangle $PQR$ respectively,such that $PQ = 12.5 \, cm$,$PA = 5 \, cm$,$BR = 6 \, cm$,and $PB = 4 \, cm$. Is $AB \parallel QR$? Give reasons for your answer.
(A) Given: $PQ = 12.5 \, cm$,$PA = 5 \, cm$,$BR = 6 \, cm$,and $PB = 4 \, cm$.
First,calculate the length of $AQ$:
$AQ = PQ - PA = 12.5 - 5 = 7.5 \, cm$.
Now,calculate the ratios:
$\frac{PA}{AQ} = \frac{5}{7.5} = \frac{50}{75} = \frac{2}{3}$ ......$(i)$
$\frac{PB}{BR} = \frac{4}{6} = \frac{2}{3}$ ......$(ii)$
From equations $(i)$ and $(ii)$,we observe that:
$\frac{PA}{AQ} = \frac{PB}{BR}$
According to the converse of the Basic Proportionality Theorem (Thales Theorem),if a line divides any two sides of a triangle in the same ratio,then the line is parallel to the third side.
Therefore,$AB \parallel QR$.
First,calculate the length of $AQ$:
$AQ = PQ - PA = 12.5 - 5 = 7.5 \, cm$.
Now,calculate the ratios:
$\frac{PA}{AQ} = \frac{5}{7.5} = \frac{50}{75} = \frac{2}{3}$ ......$(i)$
$\frac{PB}{BR} = \frac{4}{6} = \frac{2}{3}$ ......$(ii)$
From equations $(i)$ and $(ii)$,we observe that:
$\frac{PA}{AQ} = \frac{PB}{BR}$
According to the converse of the Basic Proportionality Theorem (Thales Theorem),if a line divides any two sides of a triangle in the same ratio,then the line is parallel to the third side.
Therefore,$AB \parallel QR$.
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