In an obtuse triangle ABC AB = BC, AB = 15, the height of CH is 9, find the cosine of the angle ABC

Since, by condition, AB = BC, then BC = 15 cm.

In a right-angled triangle СВН, according to the Pythagorean theorem, we determine the length of the ВН leg.

BH ^ 2 = BC ^ 2 – CH ^ 2 = 225 – 81 = 144.

BH = 12 cm.

Then AH = 12 + 15 = 27 cm.

In a right-angled triangle ACН, according to the Pythagorean theorem: AC ^ 2 = CH ^ 2 + AH ^ 2 = 81 + 729 = 810.

Then in the triangle ABC, by the cosine theorem:

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * CosABC.

810 = 225 + 225 – 2 * 15 * 15 * CosABC.

450 * CosABC = 450 – 810 = -360.

CosABC = -360 / 450 = -0.8.

Answer: CosABC is – 0.8.