Final answer:
The length of segment DC is 34.286.
Step-by-step explanation:
The ratio of DE to DF is 5:7. We know that triangle ABC is isosceles with AB=AC. Therefore, DE and DF are corresponding parts of congruent triangles, so they must have the same ratio as AB and AC. We can set up the proportion:
AB/AC = DE/DF
5/7 = AB/AC
Since AB=AC, we can substitute AB for AC in the proportion:
5/7 = 1/AC
Now, we can solve for AC:
7/5 = 1/AC
Cross-multiplying, we get:
7AC = 5
AC = 5/7
Therefore, the length of segment DC is 5/7 times the length of BC. Since BC = 48, DC = (5/7)*48 = 34.286.