In triangle ABC, the AC side is 26, and the medians drawn from the vertices A and C are 36

In triangle ABC, the AC side is 26, and the medians drawn from the vertices A and C are 36 and 15, respectively. Find the third median.

Let the point M be the point of intersection of the medians.

The medians of the triangle intersect at point M and divide it in a ratio of 2: 1, counting from the apex of the angle.

AM = 36 * 2/3 = 12 * 2 = 24,
CM = 15 * 2/3 = 5 * 2 = 10.

Consider the aspect ratio of triangle AMC.

AM ^ 2 + CM ^ 2 = 24 ^ 2 + 10 ^ 2 = 576 + 100 = 676 = 262 = AC2.

The Pythagorean theorem for a right-angled triangle is fulfilled, therefore, at the vertex M, the angle of the straight line.

The median in a right-angled triangle, drawn to the hypotenuse, is equal to half of the hypotenuse.
This means that one third of the desired median is 26/2 = 13.

Let’s find the length of the unknown median.

13 * 3 = 39.

Answer: 39.