Tangents to the circle are drawn through the ends of a chord equal to the radius

| September 11, 2021 | education

Tangents to the circle are drawn through the ends of a chord equal to the radius. Find the angles at the intersection of a tangent and a line containing a given chord

We will also draw the points O, the center of the circle, the radii OA and OB.

Since, according to the condition, the length of the chord AB is equal to the radius of the circle, then AB = OA = OB = R, and therefore, the triangle AOB is equilateral and all of its internal angles are 60.

By the property of tangents, the angle between the tangent and the radius of the circle drawn to the tangent point is 90.

Then, in the quadrangle ОАСВ the angle АСВ = 360 – АВВ – ОАС – ОВС = 360 – 60 – 90 – 90 = 120.

By the property of a tangent drawn from one point, the segment AC = BC, and then the triangle ACB is isosceles, and the angle CAB = CBA = (180 – 120) / 2 = 30. Then the angle DAM = 180 – CAB = 150 – 30 = 50.

Answer: a) the angles are 30 and 150.



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