If two sides ^@AB^@ and ^@BC,^@ and the median ^@AD^@ of ^@ \tri | Math Question and Answer

Answer:

Step by Step Explanation:

We are given that $A B = P Q, B C = Q R, and A D = P M .$

Let us now draw the triangles and mark the equal sides and medians.
We need to prove that $\triangle ABC \cong \triangle PQR.$
It is given that $\begin{aligned} &BC = QR \\ \implies &\dfrac{1}{2} BC = \dfrac{1}{2} QR \\ \implies & BD = QM && \ldots (1) && \text { [As the median from a vertex of a triangle bisects the opposite side.] } \end{aligned}$ Now, in $\triangle ABD$ and $\triangle PQM,$ we have $\begin{aligned} & AD = PM && [\text{Given}] \\ & AB = PQ && [\text{Given}] \\ & BD = QM && [\text{From (1)}] \\ \therefore \space &\triangle ABD \cong \triangle PQM && [\text{By SSS criterion}] \end{aligned}$
As corresponding parts of congruent triangles are equal, we have $\angle B = \angle Q \space \space \space \ldots (2)$
In $\triangle ABC$ and $\triangle PQR,$ we have $\begin{aligned} & BC = QR && [\text{Given}] \\ & AB = PQ && [\text{Given}] \\ & \angle B = \angle Q && [\text{From (2)}] \\ \therefore \space & \triangle ABC \cong \triangle PQR && [\text{By SAS criterion}] \end{aligned}$
Hence Proved.

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