Mathematics
Show that the line segment joining the points of contact of two parallel tangents passes through the centre.
Answer
Let AB and CD are parallel tangents of circle with centre O.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
∠APO = 90°
⇒ ∠APO + ∠EOP = 180° (sum of adjacent interior angles)
⇒ ∠EOP = 180° - 90°
⇒ ∠EOP = 90°
Similarly,
⇒ ∠EOQ + ∠CQO = 180° (sum of adjacent interior angles)
⇒ ∠EOQ = 180° - 90°
⇒ ∠EOQ = 90°
∠EOP + ∠EOQ = 90° + 90° = 180°
∴ POQ is a straight line.
Hence, line segment joining the points of contact of two parallel tangents passes through the centre.
Related Questions
Show that the tangent lines at the end points of a diameter of a circle are parallel.
Prove that the tangents at the extremities of any chord make equal angles with the chord.
In the given figure, PQ is a transverse common tangent to two circles with centres A and B and of radii 5 cm and 3 cm respectively. If PQ intersects AB at C such that CP = 12 cm, calculate AB.
ΔABC is an isosceles triangle in which AB = AC, circumscribed about a circle. Prove that the base is bisected by the point of contact.
