In a circle, the distance O K from the center O to the chord AB is 3. Find the radius of the circle if the length

| August 22, 2021 | education

In a circle, the distance O K from the center O to the chord AB is 3. Find the radius of the circle if the length of the chord AB is 8.

Let a circle be given with the center at point O, its chord AB = 8 cm, and the distance from the chord to the center of the circle OK = 3 cm.Let’s find its radius.
The shortest distance from a point to a straight line is a perpendicular, which means OK_ | _AB. Let’s connect points A and B with the center of circle O. We have got a triangle AOB.
Consider the triangle AOB, AO = OB = R – as the radii of the circle, AB = 8 cm. Since OK is perpendicular to AB, this means that this is the height of the triangle AOB. We have AO = OB – an isosceles triangle, and the height of an isosceles triangle lowered to the base is its median and bisector. By the property of the median:
AK = KB = AB / 2 = 8/2 = 4 cm.
Consider a triangle AOK, it is rectangular <K = 90 °, legs OK = 3 cm, AK = 4 cm, we will find its hypotenuse according to the Pythagorean theorem:
AO = √ (AK² + OK²) = √ (4² + 3²) = √25 = 5 cm.
Answer: the radius of the circle is 5 cm.



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