Final answer:
In similar triangles, the sides and medians are proportional. By considering the medians as segments within the smaller triangles formed, we derive proportions that show the medians are as proportional as the sides of similar triangles.
Step-by-step explanation:
To prove that in similar triangles, the ratio of their corresponding sides is equal to the ratio of their corresponding medians, we need to understand the definition of similar triangles and properties of medians in triangles. Similar triangles are those that have the same shape but are not necessarily the same size. This is because their corresponding angles are equal and their corresponding sides are proportional.
Let's consider two similar triangles, ∆ABC and ∆DEF, with corresponding sides AB/DE = BC/EF = AC/DF. The medians of a triangle connect a vertex to the midpoint of the opposite side. Thus, in ∆ABC, if G is the midpoint of BC and M is the midpoint of EF in ∆DEF, then AG and DM are medians of their respective triangles.
Since AG divides ∆ABC into two smaller triangles, ∆AGB and ∆AGC, which are similar to ∆DMF and ∆DME respectively, the sides of these triangles are also proportional. As a result, we can set up a proportion AG/DM = AB/DE. Thus, the medians of similar triangles are proportional to the corresponding sides.