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 concept of similarity of triangles:
Two triangles are said to be similar if their corresponding sides are proportional, and their corresponding angles are equal.
Given that sides $AB$, $BC$, and the median $AD$ of $△ABC$ are respectively proportional to sides $PQ$, $PR$, and median $PM$ of $△PQR$, we need to prove that $△ABC ∼ △PQR$.

Step 2: Proportionality condition:
We are given that:
\[ \frac{AB}{PQ} = \frac{BC}{PR} = \frac{AD}{PM} = k \] This shows that the corresponding sides of the two triangles are proportional.

Step 3: Using the properties of medians:
In two similar triangles, the corresponding medians are also proportional. Since $AD$ is the median of $△ABC$ and $PM$ is the median of $△PQR$, we know that the medians also satisfy the proportionality condition: \[ \frac{AD}{PM} = k \]
This confirms that the corresponding medians are proportional.

Step 4: Conclusion - Similarity of triangles:
Since the corresponding sides and medians of $△ABC$ and $△PQR$ are proportional, by the criteria of similarity (SSS similarity criterion), we conclude that:
\[ △ABC ∼ △PQR \] Thus, we have shown that $△ABC$ is similar to $△PQR$.