In triangle ABC, sides AB = BC = 5m, AC = 8m, median AK and bisector BH intersect at point M. Find BM and AK.

The bisector BH of an isosceles triangle ABC will also be the height and median, then AH = CH = AC / 2 = 8/2 = 4 m, and triangles ABH and CBH are rectangular.

We calculate the length of the leg BH of the right-angled triangle BCH.

BH ^ 2 = BC ^ 2 – CH ^ 2 = 25 = 16 = 9.

BH = 9 m.

Since BH and AK are medians, at point M they are divided in a ratio of 2/1 starting from the top.

BM = 2 * MH.

BM + MH = 3 * MH = 9 m.

MH = 9/3 = 3 m.

BM = 2 * 3 = 6 m.

Then from the right-angled triangle AMH, AM ^ 2 = AH ^ 2 + MH ^ 2 = 16 + 9 = 25.

AM = 5 m, then MK = 5/2 = 2.5 m.

AK = AM + MK = 5 + 2.5 = 7.5 m.

Answer: The length of the BM segment is 6 m, the length of the AK is 7.5 m.