The height of the triangle splits its base into two segments with lengths 8 and 9

The height of the triangle splits its base into two segments with lengths 8 and 9. Find the length of this height if you know that another height divides it in half.

The triangle will be denoted by ABC. Let the height dividing the base be BT and the second height CX. We denote the point at which they intersect by M. By the condition BM = MT. Right-angled triangles MBX and ABT are similar (similarity mark is the first). Triangles MCT and MBX are also similar (similarity feature first). So triangles ABT and MCT are also similar.

We get AT / MT = BT / TC;
8 / MT = BT / 9;
8 / MT = 2 * MT / 9;
8 * 9 = 2 * MT * MT;
36 = MT * MT;
MT = 6;
BT = 2 * MT = 12.

Answer: height length BT = 12.