If AD and PM are medians of ABC and PQR, respectively where ABC ~ PQR, prove that AB AD PQ PM 6.5 Summary In this chapter you have studied the following points :
Step-by-Step Solution
Key Concept: In two similar triangles, the line joining a vertex to the midpoint of the opposite side (a median) corresponds to the median of the other triangle. Hence the ratio of corresponding sides equals the ratio of the corresponding medians.
1. Given \(\triangle ABC \sim \triangle PQR\).\
Therefore, corresponding sides are in proportion:
$$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}\tag{1}$$
2. Mid‑points: Since \(AD\) is a median of \(\triangle ABC\), \(D\) is the midpoint of \(BC\). Similarly, \(PM\) is a median of \(\triangle PQR\), so \(M\) is the midpoint of \(QR\).
3. Correspondence of mid‑points: Under the similarity \(\triangle ABC \sim \triangle PQR\), the side \(BC\) corresponds to \(QR\). Hence the midpoint of \(BC\) (point \(D\)) corresponds to the midpoint of \(QR\) (point \(M\)).
4. Corresponding medians: The segment joining a vertex to the midpoint of the opposite side is uniquely determined by the two vertices involved. Consequently, the median \(AD\) of \(\triangle ABC\) corresponds to the median \(PM\) of \(\triangle PQR\).
5. Proportionality of corresponding medians: Because corresponding elements of similar figures are in the same ratio, we have
$$\frac{AD}{PM}=\frac{AB}{PQ}\tag{2}$$
(the ratio of a side to its median is the same as the ratio of the corresponding side to its median in the similar triangle).
6. Re‑arranging (2) gives the required result:
$$\frac{AB}{AD}=\frac{PQ}{PM}$$
Thus, the ratio of a side to its median in \(\triangle ABC\) equals the ratio of the corresponding side to its median in \(\triangle PQR\).
Correct Answer: \(\displaystyle \frac{AB}{AD}=\frac{PQ}{PM}\)