Introduction to tangent and secant of a circle:
A tangent of a circle is a line drawn from a point passing through the circle just at one point and secant is a line passing through two points of a circle. In case, a tangent and a secant are drawn from the same point outside the circle, we can find an interesting relation. Similar is the case when two secants are drawn from the same point outside the circle.
Let us study these two situations.
Tangent and Secant of a Circle – a Tangent and a Secant
Look at the above diagram.
A tangent OT is drawn from O touching the circle at P. From the same point O, a secant OS is drawn passing through the points Q and R on the circle. Join PR and PQ.
The angle OPQ and ORP are congruent as both of them are subtended by the intercepted arc PQ.
Angle POR is subtended both by the chords PQ and PR at O. Therefore the triangles OPQ and ORP are similar.
Applying the rule of similarity,
OP/OR = OQ/OP
or, OP2 = OQ*OR
Tangent and Secant of a Circle – Two Secants
Look at the above diagram.
Two secants are drawn from the same point O to the circle. One secant passes through the points A and B on the circle and the other passes through the points C and D on the circle. Join BC and DA.
The angles BAD and BCD are subtended by the same arc BD on the circumference on the circle. Hence these two angles are congruent. The angle at O is common to the triangle AOD and BOC. Therefore, the triangles AOD and BOC are similar.
Applying the rule of similarity,
OC/OA = OB/OD
or, OA*OB = OC*OD
The two properties derived are very useful in solving problems related circles with tangents and secants.
A tangent of a circle is a line drawn from a point passing through the circle just at one point and secant is a line passing through two points of a circle. In case, a tangent and a secant are drawn from the same point outside the circle, we can find an interesting relation. Similar is the case when two secants are drawn from the same point outside the circle.
Let us study these two situations.
Tangent and Secant of a Circle – a Tangent and a Secant
Look at the above diagram.
A tangent OT is drawn from O touching the circle at P. From the same point O, a secant OS is drawn passing through the points Q and R on the circle. Join PR and PQ.
The angle OPQ and ORP are congruent as both of them are subtended by the intercepted arc PQ.
Angle POR is subtended both by the chords PQ and PR at O. Therefore the triangles OPQ and ORP are similar.
Applying the rule of similarity,
OP/OR = OQ/OP
or, OP2 = OQ*OR
Tangent and Secant of a Circle – Two Secants
Look at the above diagram.
Two secants are drawn from the same point O to the circle. One secant passes through the points A and B on the circle and the other passes through the points C and D on the circle. Join BC and DA.
The angles BAD and BCD are subtended by the same arc BD on the circumference on the circle. Hence these two angles are congruent. The angle at O is common to the triangle AOD and BOC. Therefore, the triangles AOD and BOC are similar.
Applying the rule of similarity,
OC/OA = OB/OD
or, OA*OB = OC*OD
The two properties derived are very useful in solving problems related circles with tangents and secants.
