# The area of the triangle ABC is 60√2. Find AC if AB side is 11 and it is more than half of AC side and BM median is 10.

Since BM is the median of triangle ABC, she divided it into two equal triangles.

Sawm = Svcm = Savs / 2 = 60 * √2 / 2 = 30 * √2 cm2.

In triangle ABM, we will construct the height of MH to the side AB.

Savs = 30 * √2 = AB * MH / 2.

МН = 2 * Saс / AB = 2 * 30 * √2 / 11 = 60 * √2 / 11 cm.

In a right-angled triangle BHM, according to the Pythagorean theorem, BH ^ 2 = BM ^ 2 – MH ^ 2 = 100 – 7200/121 = (12100 – 7200) / 121 = 4900/121.

BH = 70/11 cm.

Then AH = AB – BH = 11 – 70/11 = 51/11 cm.

In a right-angled triangle AHM, according to the Pythagorean theorem, AM ^ 2 = AH ^ 2 + MH ^ 2 = (2601/121) + (7200/121) = 9801/121 = 81.

AM = 9 cm.

Since BM is the median, then AC = 2 * AM = 2 * 9 = 18 cm.

Answer: The length of the speaker side is 18 cm.

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