Point C divides chord AB into segments 12 cm and 14 cm long. Find the radius of the circle

| March 5, 2021 | education

Point C divides chord AB into segments 12 cm and 14 cm long. Find the radius of the circle if the distance from the center of the circle to point C is 11 cm.

Let us determine the length of the chord AB.

AB = AC + BC = 12 + 14 = 28 cm.

Let’s draw the radii OA and OB from the center of the circle. Since ОА = ОВ = R, the triangle AOB is isosceles.

Let’s draw the height OH of the triangle AOB, which will also be the median of the triangle.

Then AH = BH = AB / 2 = 28/2 = 14 cm.

Segment CH = AН – AC = 14 – 12 = 2 cm.

In a right-angled triangle СOН, according to the Pythagorean theorem, we determine the length of the leg OH.

OH ^ 2 = OC ^ 2 – CH ^ 2 = 11 ^ 2 – 2 ^ 2 = 121 – 4 = 117.

In a right-angled triangle AOН, by the Pythagorean theorem, we determine the length of the hypotenuse OA.

OA ^ 2 = AH ^ 2 + OH ^ 2 = 196 + 117 = 313.

ОА = R = √313 cm.

Answer: The radius of the circle is √313 cm.



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