The ratio of similar triangle,Ratio of median=ratio of sidesHence,Ratio of median = 5:7

Introduction:
Let's consider two similar triangles, Triangle ABC and Triangle DEF. We are given that the corresponding medians of these triangles are in the ratio 5:7. We need to find the ratio of their sides.

Understanding Similar Triangles:
Similar triangles have the same shape but may have different sizes. The corresponding angles of similar triangles are equal, and the corresponding sides are proportional to each other. This means that if we have two similar triangles, we can compare the lengths of their corresponding sides using ratios.

Medians of a Triangle:
The median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In a triangle, the medians intersect at a point called the centroid. The centroid divides each median into two segments, with the ratio of the lengths of these segments being 2:1.

Ratio of Medians:
In our given problem, the medians of Triangle ABC and Triangle DEF are in the ratio 5:7. This means that the lengths of the medians are in the ratio 5:7. Let's assume the lengths of the medians of Triangle ABC and Triangle DEF are 5x and 7x respectively.

Ratio of Sides:
As mentioned earlier, the centroid divides each median into two segments with a ratio of 2:1. Therefore, the ratio of the lengths of the sides of Triangle ABC and Triangle DEF will be the same as the ratio of the lengths of the medians.

Let's consider the sides of Triangle ABC. The ratio of the lengths of the medians is 5:7, so the ratio of the lengths of the sides will also be 5:7. Therefore, the ratio of the sides of Triangle ABC is 5x:7x.

Similarly, for Triangle DEF, the ratio of the lengths of the sides will be 5x:7x.

Conclusion:
In summary, if the corresponding medians of two similar triangles are in the ratio 5:7, then the ratio of their sides will also be 5:7. This is because the centroid divides each median into two segments with a ratio of 2:1, which is the same as the ratio of the lengths of the sides.