How do I solve for BG and BF? I know how to solve for FD, AD, and AG.

Question

589.Triangle ABC has AB = 12 = AC and angle A is 120 degrees. Let F and D be the midpoints of sides AC and BC, respectively, and G be the intersection of segments AD and BF. Find the lengths FD, AD, AG, BG, and BF.

Answer 1

Segments AD and BF are medians intersecting at point G which is the centroid of the triangle.

You can find BG by using the Pythagorean Theorem because triangle BGD is a right triangle and GD = 2, BD = 6√3. So BG = √112 = 4√7.

The centroid (G) is 2/3 of the way along the median starting at vertex B. So, (2/3) BF = 4√7 => BF = (3/2)4√7=6√7.

Here is a Desmos graph that validates the above:

desmos.com/calculator/vkyn7f63yh