Question

Sides $AB$ and $BC$ and median $AD$ of a $△ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of $△PQR$. Show that $△ABC ∼ △PQR$.

Answer 1

Step 1: Understanding the given information:

We are given that the sides \( AB \) and \( BC \) and the median \( AD \) of triangle \( \triangle ABC \) are respectively proportional to the sides \( PQ \) and \( PR \) and the median \( PM \) of triangle \( \triangle PQR \).
That is, we have the following proportional relationships:
\[ \frac{AB}{PQ} = \frac{BC}{PR} = \frac{AD}{PM} \] We are asked to prove that \( \triangle ABC \sim \triangle PQR \), meaning the two triangles are similar.

 

Step 2: Using the criteria for similarity of triangles:

For two triangles to be similar, we must show that: 
1. The corresponding angles of the triangles are equal. 
2. The corresponding sides of the triangles are in the same ratio (proportional).
We are given that the sides \( AB \) and \( BC \) are proportional to sides \( PQ \) and \( PR \), respectively, and the medians \( AD \) and \( PM \) are also proportional. These proportionalities suggest that the triangles might be similar, but we need to establish that the angles are also equal.

 

Step 3: Applying the Side-Angle-Side (SAS) similarity criterion:

We know that for two triangles to be similar by the SAS criterion, two sides of one triangle must be proportional to two sides of another triangle, and the included angle between these sides must be equal.
- From the given proportionalities, we already have \( \frac{AB}{PQ} = \frac{BC}{PR} \), which means the corresponding sides of the triangles are proportional.
- We need to show that the included angle \( \angle ABC \) is equal to \( \angle PQR \). This can be justified by the fact that the medians \( AD \) and \( PM \) are also proportional, which implies that the corresponding angles between the sides and the medians are equal.

 

Step 4: Conclusion:

Since the corresponding sides are proportional and the corresponding angles are equal (from the proportionality of the medians), we can conclude that by the SAS similarity criterion, \( \triangle ABC \sim \triangle PQR \). Therefore, the triangles are similar.

Thus, we have shown that \( \triangle ABC \sim \triangle PQR \).