Sides AB and AC and median AM of a △ ABC are proportional to sides DE and DF and median DN of another △ DEF . Show…
CBSE Class 10 Maths PYQ · Triangles · Similarity with Triangles · 5 Marks · March 2023 · Standard
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1005 Marks · March 2023 · Standard
Sides $AB$ and $AC$ and median $AM$ of a $\triangle ABC$ are proportional to sides $DE$ and $DF$ and median $DN$ of another $\triangle DEF$. Show that $\triangle ABC \sim \triangle DEF$
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1 mark for figure Extend $AM$ to $A'$ so that $AM = A'M$ and $DN$ to $D'$ so that $DN = D'N$. Join $A'C$ and $D'F$. $$\begin{aligned}& \triangle AMB \sim \triangle A'MC \\ & \Rightarrow AB = A'C\end{aligned}$$. Similarly, $$\begin{aligned}& DE = D'F \\ & \text{Given } \frac{AB}{DE} = \frac{AC}{DF} = \frac{AM}{DN} \\ & \Rightarrow \frac{AC}{DF} = \frac{A'C}{D'F} = \frac{AA'/2}{DD'/2} \\ & \therefore \triangle AA'C \sim \triangle DD'F \\ & \therefore \angle 1 = \angle 2 \\ & \text{Similarly, } \angle 3 = \angle 4 \\ & \Rightarrow \angle 1 + \angle 3 = \angle A = \angle 2 + \angle 4 = \angle D \\ & \text{Hence } A ABC \sim \triangle DEF \text{ (SAS)}\end{aligned}$$
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