Answer:
BC = 62
Explanation:
There are 3 triangles total in the figure shown, one large triangle, and two smaller triangles created by the segment BC. Now, look at triangle ACD and see that is a scaled version of common right triangle.
Some common right triangles are 3-4-5, 5-12-13, and 7-24-25. You may be able to see a pattern here where the square of the smallest side length is equal to the sum of the other two side lengths. Triangle ACD happens to be a scaled form of a 5-12-13 triangle. The triangle is scaled by a factor of 3. So the respective side length of AD is 12 * 3 = 36.
Now look at triangle ABD. We have two of the three side lengths.
AD = 36
AB = 85
We want to find BC, so we will use the pythagorean theorem, with (x + 15) to represent the length of BD.
35² + (x + 15)² = 85²
1225 + (x + 15)² = 7225
(x + 15)² = 6000
x + 15 = ±√6000
x = -15 ± √6000
Our equation provides two answers, but the negative value doesn't make sense because length can't be negative, so use the positive length for this application.
x = -15 + √6000
x = 62.4596
Rounded to the nearest whole number, the length of BC is 62.