Tangents to a Circle

Summary

A straight line intersects a circle at one or two point. The tangent to a circle is a line that touches the circle at one point. Secant intersects the circle at two points.

Point of contact:
The point at which the straight line touches the circle is called the point of contact or point of tangency.

Some properties of tangents to a circle:

  • Infinite number of tangents can be drawn to a circle but only one tangent can be drawn at any given point on a circle.
  • From an external point to can draw two tangents of equal length.
  • The radius of the circle is perpendicular to the tangent at its point of contact and the tangents drawn at the extremities of the diameter of a circle are parallel.


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Theorem:
The tangent at any point on a circle is perpendicular to the radius drawn to the point of contact.



Given: A tangent AB with point of contact P.
To prove: OP ⊥ AB
Proof:
Consider point C on AB other than P.
C must lie outside the circle. (∵ A tangent can have only one point of contact with the circle)
OC > OP (∵ C lies outside the circle)
This is true for all positions of C on AB.
Thus, OP is the shortest distance between point P and line segment AB.
Hence, OP ⊥ AB.

Theorem:
Tangents drawn to a circle from an external point are equal in length.



Given: Two tangents AB and AC from an external point A to points B and C on a circle.
To prove: AB = AC
Construction: Join OA, OB and OC.
Proof:
In triangles OAB and OAC,
∠OBA = 90o (Radius OB ⊥ Tangent AB at B)
∠OCA = 90o (Radius OC ⊥ Tangent AC at C)
In triangles OBA and OCA,
∠OBA = ∠OCA = 90o
OB = OC (Radii of the same circle)
OA = OA (Common side)
Thus, ΔOBA ≅ ΔOCA (RHS congruence rule)
Hence, AB = AC (Corresponding sides of congruent triangles)
The tangents drawn at the extremities of the diameter of a circle are parallel.


 

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