Question

Sides AB and BC and median AD of a triangle ABC are proportional to sides PQ and PR and median PM of the triangle PQR. Prove triangle $ABC\sim PQR$ .

Answer 1

Hint: According to the given question, we need to show the similarity between two triangles ABC and PQR. So, for this we need to use various similarity properties and therefore using various similarity criterion we will prove that the given two triangles are similar.

Complete step by step answer:
According to the question, we are given two triangles and for triangle ABC we have AB and BC and AC as three sides and AD be the median of the two triangles.
EFG=PQR in figure and PM =EH.
Let the two triangles be as shown below:

So, now we know that AB, BC and AD of ABC are proportional to the sides PQ, QR and median PM of triangle PQR.
This implies that $\dfrac{AB}{PQ}=\dfrac{BC}{QR}=\dfrac{AD}{PM}$
We need to prove: $ABC\sim PQR$.
Proof: $\dfrac{AB}{PQ}=\dfrac{BC}{QR}=\dfrac{AD}{PM}$
$\dfrac{AB}{PQ}=\dfrac{BC}{QR}=\dfrac{AD}{PM}$in this D is the mid-point of BC and M is the mid-point of QR.
Triangles \[ABD\sim PQM\] using SSS criterion.
Therefore, $\angle ABD=\angle PQM$ as we know that corresponding angles of two similar triangles are equal.
Now, $\angle ABC=\angle PQR$
In triangles ABC and PQR
$\dfrac{AB}{PQ}=\dfrac{BC}{QR}$
Now, $\angle ABC=\angle PQR$
So, from above two we have $ABC\sim PQR$using SAS criterion.
Therefore, we have proved that the two triangles ABC and PQR are similar.

Note: We need to be little careful while proving the two triangles as similar triangles as some of the properties of proving congruence of two triangles is also same, so we need to use them wisely and be very clear with the concept of the congruence and similarity of two triangles.