# The lengths of the two sides of the triangle are 27 and 29. The length of the median

The lengths of the two sides of the triangle are 27 and 29. The length of the median to the third side is 26. Find the height of the triangle to the side with length 27.

Let’s designate this triangle ABC.
AB = 27, BC = 29, BM – median to the AC side, BM = 26. CH – height to AB side.
To find the third side of the AC, we use the formula for the median of a triangle.
BM = 1/2 * √ (2 * AB² + 2 * BC² – AC²)
2 * BM = √ (2 * AB² + 2 * BC² – AC²)
4 * BM² = 2 * AB² + 2 * BC² – AC²
AC² = 2 * AB² + 2 * BC² – 4 * BM² = 2 * 729 + 2 * 841 – 4 * 676 = 1458 + 1682 – 2704 = 436
AC = √436.
The CH height divides the AB side into segments:
BH = x, AH = 27 – x.
In a right-angled triangle ВСН, we write down the cathetus СН according to the Pythagorean theorem:
CH² = BC² – BH² = 841 – x².
Similarly, we write the same CH leg in the ASN right-angled triangle:
CH² = AC² – AH² = 436 – (27 – x) ² = 436 – 729 + 54x – x².
We equate and get the equation:
841 – x² = 436 – 729 + 54x – x²
54x = 1134
x = 21
CH² = 841 – x² = 841 – 441 = 400 → CH = 20.