Question

If AP and DQ are medians of triangles ABC and DEF respectively, where \( \triangle ABC \sim \triangle DEF \), prove that: \[ \frac{AB}{DE} = \frac{AP}{DQ} \]

Hint:

In similar triangles, corresponding sides and corresponding medians are proportional.

Answer 1

Step 1: Understand the given information.
We are given two triangles \( \triangle ABC \) and \( \triangle DEF \), with medians \( AP \) and \( DQ \) respectively. We also know that the two triangles are similar, i.e., \[ \triangle ABC \sim \triangle DEF. \]
Step 2: Use the property of similar triangles.
From the property of similar triangles, we know that the corresponding sides of similar triangles are proportional. Therefore, we have: \[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}. \]
Step 3: Use the property of medians in similar triangles.
In a triangle, the medians divide the triangle into two smaller triangles of equal area. Since \( \triangle ABC \sim \triangle DEF \), the corresponding medians \( AP \) and \( DQ \) must be proportional to the corresponding sides \( AB \) and \( DE \). Thus, we can write the proportion: \[ \frac{AB}{DE} = \frac{AP}{DQ}. \]
Step 4: Conclusion.
Hence, we have proved that: \[ \frac{AB}{DE} = \frac{AP}{DQ}. \]