The height of the triangle, equal to 10 cm, divides the base into two segments equal to 10 cm and 4 cm.

| September 11, 2021 | education

The height of the triangle, equal to 10 cm, divides the base into two segments equal to 10 cm and 4 cm. Find the median drawn to the smaller of the two sides.

In a right-angled triangle ABH, according to the Pythagorean theorem, AB ^ 2 = BH ^ 2 + AH ^ 2 = 100 + 16 = 116.

AB = 2 * √29 cm.

In a right-angled triangle СBН, according to the Pythagorean theorem, CB ^ 2 = BH ^ 2 + CH ^ 2 = 100 + 100 = 200.

AB = 2 * √10 cm.

AC = AH + CH = 4 + 10 = 14 cm.

In the triangle ABC, according to the cosine theorem: BC ^ 2 = AC ^ 2 + AB ^ 2 – 2 * AC * AB * CosA.

200 = 196 + 116 – 2 * 14 * 2 * √29 * CosA.

56 * √29 * CosA = 112.

CosA = 2 / √29.

Since CM is the median, then AM = AB / 2 = √29.

Then in the triangle ACM, by the cosine theorem: CM ^ 2 = AM ^ 2 + AC ^ 2 – 2 * AM * CM * CosA =

29 + 196 – 2 * √29 * 14 * 2 / √29 = 169.

CM = 13 cm.

Answer: The median length is 13 cm.



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