Question

Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.

Answer 1

Given : CD and EF are two parallel tangents at the points A and B of a circle with center O.

To prove : AOB is a diameter of the circle

Construction : Join OA and OB

Draw OG | | CD

Proof : OG | | CD and AO cuts them .

⇒ ${90}^{\circ }$ + GOA = ${180}^{\circ }$ [ OA is perpendicular to CD ]

⇒ GOA = ${90}^{\circ }$)

Similarly, GOB = ${90}^{\circ }$;

Therefore, GOA + GOB = (${90}^{\circ }$ + ${90}^{\circ }$) = ${180}^{\circ }$)

=> AOB is a straight line

Hence, AOB is a diameter of the circle with center O.