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.

univerkov | July 30, 2021 | education

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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