Question

Sides AB and BC and median AD of a \(\triangle ABC\) are respectively proportional to sides PQ and QR and median PM of another \(\triangle PQR\). Then prove that \(\triangle ABC \sim \triangle PQR\).

Hint:

When a problem involves medians and proportionality, try to use the fact that a median divides a side into two equal halves. This often allows you to establish similarity for smaller triangles, which then helps prove the similarity of the larger triangles.

Answer 1

Step 1: Understanding the Concept:
To prove that \(\triangle ABC \sim \triangle PQR\), we can use the SAS (Side-Angle-Side) similarity criterion. The strategy is to first prove that two smaller triangles formed by the medians are similar (\(\triangle ABD \sim \triangle PQM\)) to establish the equality of \(\angle B\) and \(\angle Q\). A common typo in this problem is listing PR instead of QR; we will assume the proportion is with QR.

Step 2: Detailed Explanation:
We are given: \[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AD}{PM} \] where AD and PM are medians. This means D is the midpoint of BC and M is the midpoint of QR.
So, \(BC = 2BD\) and \(QR = 2QM\).
Substitute this into the given proportion: \[ \frac{AB}{PQ} = \frac{2BD}{2QM} = \frac{AD}{PM} \] \[ \implies \frac{AB}{PQ} = \frac{BD}{QM} = \frac{AD}{PM} \] Now consider \(\triangle ABD\) and \(\triangle PQM\). Since all three corresponding sides are in proportion, by the SSS (Side-Side-Side) similarity criterion, we have: \[ \triangle ABD \sim \triangle PQM \] Because the triangles are similar, their corresponding angles are equal. \[ \therefore \angle B = \angle Q \] Now, let's consider the original triangles, \(\triangle ABC\) and \(\triangle PQR\). We have: \begin{enumerate} \item \(\frac{AB}{PQ} = \frac{BC}{QR}\) (Given). \item \(\angle B = \angle Q\) (Proved above). \end{enumerate} By the SAS (Side-Angle-Side) similarity criterion, we can conclude that: \[ \triangle ABC \sim \triangle PQR \]

Step 3: Final Answer:
Hence, it is proved that \(\triangle ABC\) is similar to \(\triangle PQR\).