In a right-angled triangle ABC, legs AB = 8, BC = 6. A point K is marked on the hypotenuse AC

| June 18, 2021 | education

In a right-angled triangle ABC, legs AB = 8, BC = 6. A point K is marked on the hypotenuse AC so that the triangle ABK is isosceles. Find the radius of the circle around the ABK triangle.

1. The radius of the circumscribed circle is found by the formula for which we need the lengths of the sides and the area of ​​the triangle ABK. The area is found by the formula, for which we need the semiperimeter of the triangle ABK So, you need to find: 1) sides AB, AK, BK; 2) p = (AB + AK + BK): 2; 3) SABK; 4) RABK.

2. The square of the hypotenuse is equal to the sum of the squares of the legs, hence AC ^ 2 = AB ^ 2 + BC ^ 2 = 8 ^ 2 + 6 ^ 2 = 100, hence AC = 10.

3. Since ABK is an isosceles triangle, therefore AK = BK. Based on the properties of a right-angled triangle, we understand that BK is the median, which means that BK = AK = AC: 2 = 5. So, we have all three sides: AK = 5, BK = 5, AB = 8.

4.p = (AB + AK + BK): 2 = (8 + 5 + 5): 2 = 9.

5.by the formula we find S = 12.

6. by the formula we find R = 4 1/3.

Answer: R = 4 1/3.



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