In a right-angled triangle ABC, the angle is C = 90 degrees, AC = 6, CB = 8. The median CM is drawn

| May 6, 2021 | education

In a right-angled triangle ABC, the angle is C = 90 degrees, AC = 6, CB = 8. The median CM is drawn from the vertex of the right angle. A circle with center at point O is inscribed in triangle ABC. Find the area of triangle COM.

By the Pythagorean theorem, we determine the length of the hypotenuse AB.

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

AB = 10 cm.

Determine the radius of the inscribed circle. R = OH = OK = (BC + AC – AB) / 2 = (8 + 6 – 10) / 2 = 4/2 = 2 cm.

Since CM is the median, then point M is the middle of side AB, then AM = AB / 2 = 10/2 = 5 cm.

The segment PM is the middle line of the triangle ABC, then PM = BC / 2 = 8/2 = 4 cm.

Determine the area of ​​the AFM triangle. Sasm = AC * PM / 2 = 6 * 4/2 = 12 cm2.

Determine the area of ​​the triangle AOC. Saos = AC * OK / 2 = 6 * 2/2 = 6 cm2.

Determine the area of ​​the triangle AOM. Saom = AM * OH / 2 = 5 * 2/2 = 5 cm2.

Then Scom = Sasm – Saos – Saom = 12 – 6 – 5 = 1 cm2.



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