Using Basic proportionality theorem, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

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Solution

Consider the given figure in which $l$ is a line drawn through the mid-point $P$ of line segment $A B$ meeting $A C$ at $Q$ , such that $P Q | | B C$ .
Using basic proportionality theorem, we obtain
$\frac{A Q}{Q C} = \frac{A P}{P B}$
$\Rightarrow \frac{A Q}{Q C} = \frac{1}{1}$ (Since, $P$ is the mid-point of $A B$ , so $A P = P B$ )
$\Rightarrow A Q = Q C$
Hence, $Q$ is the mid-point of $A C$ .

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Using Basic proportionality theorem, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

Consider the given figure in which l is a line drawn through the mid-point P of line segment AB meeting AC at Q, such that PQ||BC.Using basic proportionality theorem, we obtain AQQC=APPB⇒AQQC=11 (Since, P is the mid-point of AB, so AP=PB)⇒AQ=QCHence, Q is the mid-point of AC.