In a triangle with sides 3,4 and 6, the median is drawn to the larger side.

In a triangle with sides 3,4 and 6, the median is drawn to the larger side. Find the cosine of the angle formed by this median with the smaller side of the triangle.

Let’s apply the formula for determining the length of the median of a triangle.

AM ^ 2 = ((2 * b ^ 2) + (2 * c ^ 2) – (a ^ 2)) / 4.

4 * AM ^ 2 = ((2 * 4 ^ 2) + (2 * 3 ^ 2) – (6 ^ 2))

4 * AM ^ 2 = 32 + 18 – 36.

AM ^ 2 = 14/4 = 7/2.

AM = √ (7/2) = √14 / 2.

In triangle ABM we apply the cosine theorem.

MB ^ 2 = AM ^ 2 + AB ^ 2 – 2 * AM * AB * CosA.

9 = (14/4) + 9 – 2 * (√14 / 2) * 3 * CosA.

6 * √14 / 2) * CosA = (7/2).

3 * √14 * CosA = 7/2.

CosA = 7 / (6 * √14) = 7 * √14 / 6 * 14 = √14 / 12.

Answer: The cosine of the angle is √14 / 12.