Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above, do the following:

In fig., O is the center of the two concentric circles. AB is a chord of the larger circle touching the smaller circle at C. Prove that AC = BC

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Solution

Take a point Q on XY other than P and join OQ.

The point Q must lie outside the circle.

Therefore, OQ is longer than the radius OP of the circle. That is, OQ $>$ OP.

Since this happens for every point on the line XY except the point P, OP is the shortest of all the distances of the point O to the points of XY. So OP is perpendicular to XY.

Join OC.

Since OP is a radius of smaller circle and ACB is a tangent to it at the point P, so OC perpendicular to AB.

But the perpendicular from the center to a chord, bisects the chord.

AC = CB

Hence, AB is bisected at the point C.

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In Fig.4, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC.

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In Fig.5, the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.

Q. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using above result , prove the following:
In figure , XZ touches the circle with centre O at Y. Diameter BA when produced meets XZ at X . Given

∠ B X Y = b

and

∠ A Y X = a

, then prove that

b + 2 a = 90^{\circ}

Q. In two concentric circles, prove that a chord of larger circle which is tangent to smaller circle is bisected at the point of contact.

Q. Prove that in two concentric circle, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
OR
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.

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Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above, do the following: In fig., O is the center of the two concentric circles. AB is a chord of the larger circle touching the smaller circle at C. Prove that AC = BC

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above, do the following:

In fig., O is the center of the two concentric circles. AB is a chord of the larger circle touching the smaller circle at C. Prove that AC = BC