In the triangle ABC, the median AM is 4 times less than the side AB and makes an angle of 60 ° with it. Find the angle MAC.

| January 6, 2021 | education

Since AM is the median of the triangle, then BM = CM, we will extend the median and complete the triangle to a parallelogram.
Then, since the diagonals at the intersection point are divided in half, then AD = 2 * AM = 2 * X.
In the AED triangle, one side is twice the size of the other, and one of the angles is 600. The conclusion suggests itself that the AED triangle is rectangular with an ADВ = 90, AED = 30 angle.
ВD^2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * Cos60 = 16 * X ^ 2 + 4 * X ^ 2 – 2 * 4 * X * 2 * X * 1/2 = 20 * X ^ 2 – 8 * X ^ 2 = 12 * X ^ 2.
BD^2 = BA ^ 2 – AD ^ 2 = 16 * X ^ 2 – 4 * X ^ 2 = 12 * X ^ 2.
The Pythagorean theorem works, the angle ADB = 90, then the angle DAM as a cross lying angle is equal to 90, and therefore MAC = 90.
Answer: Angle MAC = 900.



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