# In a right-angled triangle with a hypotenuse equal to 10 and one of the legs equal

In a right-angled triangle with a hypotenuse equal to 10 and one of the legs equal to 8, a bisector of a smaller angle is drawn. What is its length?

By the Pythagorean theorem, we determine the length of the BC leg.

BC ^ 2 = AC ^ 2 – AB ^ 2 = 100 – 64 = 36.

BC = 6 cm.

Let the length of the segment AK = X cm, then SK = (10 – X) cm.

By the property of the bisector, AB / AK = BC / SK.

8 / X = 6 / (10 – X).

6 * X = 80 – 8 * X.

14 * X = 80.

X = AK = 40/7.

In a right-angled triangle ABC, we define the cosine of the angle BAC.

CosBAC = AB / AC = 8/10 = 0.8.

In the triangle ABK, by the cosine theorem, we define the length of the segment BK.

VK ^ 2 = AB ^ 2 + AK ^ 2 – 2 * AB * AK * CosBAC = 64 + 1600/49 – 2 * 8 * (40/7) * 0.8 = (4736/49) – (3584/49 ) = 1152/49.

VK = 24 * √2 / 7 ≈ 4.85 cm.

Answer: The length of the median VC is 24 * √2 / 7 ≈ 4.85 cm.

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