tangent of a circle - Prove this, in two concentric circles, such that a chord of the bigger circle, that touches the smaller circle is bisected at the point of contact with the smaller circle.

Krishna (last edited 4 years ago) (Expert, Educator @Qalaxia)

Step 1: Draw the diagram according to the given instructions.

NOTE: We are given two concentric circles C_1 \text{and} C_2 with centre O and a chord AB of the larger circle C_1, touching the smaller circle C_2 at the point P.

Step 2: Prove that the chord is perpendicular to the radius of the smaller circle

NOTE: Large chord is tangent to the smaller circle so, it is perpendicular to the radius.

EXAMPLE: AB is perpendicular OP

Step 3: Join the center and the end points of the chord to form triangles

Step 4: Prove that two triangles are similar by using the theorems

NOTE: ∆OAP and ∆OBP are congruent. (since SAS theorem)

This means AP = PB. Therefore, OP is the bisector of the chord AB, as the

perpendicular from the centre bisects the chord.