The lengths of the legs of a right-angled triangle are 9 cm and 12 cm, calculate the distance between the point

| September 6, 2021 | education

The lengths of the legs of a right-angled triangle are 9 cm and 12 cm, calculate the distance between the point of intersection of the bisectors and the point of intersection of the medians of this triangle.

Through point K, the point of intersection of the medians, draw a straight line KН parallel to AC.

Right-angled triangles ABM and BНK are similar in acute angle.

By the property of the medians of the triangle, BK = 2 * KM, then K = 2 * BM / 3.

BK / BM = 2/3.

Then НK / AM = 2/3.

AM = AC / 2 = 12/2 = 6 cm.

KH = 2 * 6/3 = 4 cm.

Let us determine the radius of the circle inscribed in the ABC triangle.

By the Pythagorean theorem, BC ^ 2 = AB ^ 2 + AC ^ 2 = 91 + 144 = 225.BC = 15 cm.

R = OH = (AB + AC – BC) / 2 = (9 + 12 – 15) / 2 = 6/2 = 3 cm.

Then OK = KH – OH = 4 – 3 = 1 cm.

Answer: The distance between the intersection points is 1 cm.



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