In an isosceles triangle ABC, the M-point of intersection of the medians, it is 4 cm from the base

In an isosceles triangle ABC, the M-point of intersection of the medians, it is 4 cm from the base, at what distance is it from the apex.

Consider an isosceles triangle ABC with base AC and let M be the point of intersection of its medians.

We denote the medians of the triangle by AL, BN, CK.

Then, we have:

AK = BK, BL = CL, AN = CN.

Note that KL is parallel to AC and KL = 1/2 * AC.

Therefore, triangles AMC and KLM are similar to each other and

AM = 2 * LM, CM = 2 * KM. Similarly, we can get that:

BM = 2 * NM = 2 * 4 = 8.

We have proven that the median intersection divides each median by a ratio of 2 to 1, counting from the top.

In fact, we have nowhere used the fact that triangle ABC is isosceles.

Answer: 8 cm.