Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that ∆ ABC ~ ∆ PQR.
 

We are given that the sides AB and AC and median AD of triangle ABC are respectively proportional to the sides PQ and PR and median PM of triangle PQR. The goal is to prove that triangles ∆ABC and ∆PQR are similar.

Given:

• Sides AB and AC are proportional to sides PQ and PR respectively.
• Median AD of triangle ABC is proportional to median PM of triangle PQR.
• We are required to prove that ∆ABC ~ ∆PQR.

Proof:

We will use the SAS (Side-Angle-Side) Criterion for triangle similarity to prove the result.

Step 1: Extend the Medians

Extend the median AD of triangle ABC to a point E such that AD = DE. Similarly, extend the median PM of triangle PQR to a point N such that PM = MN. Now, we join CE and RN as shown below:

Step 2: Prove Triangles are Congruent

Now, consider triangles ∆ABD and ∆CDE:

• AD = DE (By construction)
• BD = DC (AD is the median of ∆ABC)
• ∠ADB = ∠CDE (Vertically opposite angles)

Thus, by the SAS Congruence Criterion, we have:

∆ABD ≅ ∆CDE, and hence AB = CE (By CPCT).

Similarly, consider triangles ∆PQM and ∆MNR:

• PM = MN (By construction)
• QM = MR (PM is the median of ∆PQR)
• ∠PMQ = ∠NMR (Vertically opposite angles)

By the SAS Congruence Criterion, we get:

∆PQM ≅ ∆MNR, and hence PQ = RN (By CPCT).

Step 3: Apply Proportionality

We are given the following proportions:

• AB / PQ = AC / PR = AD / PM

From our earlier results (i) and (ii), we know that:

• CE / RN = AC / PR = AD / PM (From AB = CE and PQ = RN)
• CE / RN = AC / PR = 2AD / 2PM (By construction, AD = AE and PM = PN)
• CE / RN = AC / PR = AE / PN

Therefore, using the SSS Similarity Criterion, we conclude that:

∆ACE ~ ∆PRN.

Step 4: Angle Correspondence

From ∆ACE ~ ∆PRN, we know the corresponding angles are equal. Hence:

• ∠CAE = ∠RPN
• ∠BAE = ∠QPN

Thus, adding the angles, we get:

• ∠CAE + ∠BAE = ∠RPN + ∠QPN
• ∠BAC = ∠QPR (By angle addition)

Therefore, we have proved that:

• ∠A = ∠P

Step 5: Conclusion

Finally, in triangles ∆ABC and ∆PQR, we have:

• AB / PQ = AC / PR (By proportionality)
• ∠A = ∠P (From earlier result)

Thus, by the SAS Similarity Criterion, we conclude that:

∆ABC ~ ∆PQR

This completes the proof.