The base of the triangle is 22 cm, the median and the height are drawn to this base, the lengths

The base of the triangle is 22 cm, the median and the height are drawn to this base, the lengths of which are 13 cm and 12 cm. Find the length of the larger lateral side of this triangle.

Median m, height h, drawn to the base of this triangle, as well as the projection of the median onto this base l form a right-angled triangle, from which, according to the Pythagorean theorem, we can find the projection of the median:

l ^ 2 = m ^ 2 – h ^ 2 = 13 ^ 2 – 12 ^ 2 = 169 – 144 = 25 = 5 ^ 2;

l = 5 cm.

Obviously, the projection of the larger lateral side of this triangle onto its base is equal to the sum of the lengths of the segments, one of which is half the base, and the other is the projection of the median onto the base. We find the square of the larger side as the sum of the squares of the height and the sum of these segments:

a ^ 2 = 12 ^ 2 + (11 + 5) ^ 2 = 144 + 256 = 400 = 202;

a = 20 cm – the desired large lateral side of this triangle.