Notes On Tangents to a Circle - CBSE Class 10 Maths
Circle
A circle is a collection of all points in a plane which are at a constant distance(radius) from a fixed point (centre).

Secant
A line which intersects a circle in two distinct points is called secant of the circle.

Tangent
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.

Length of the tangent
The length of the segment of the tangent from the external point and the point of contact with the circle is called
the length of the tangent from the external point to the circle.

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.


radius perpendicular to tangent, radius and tangent, diameter and tangent   tangents from external point, external point, tangents to a circle, tangents, tangents of equal length, equality of two tangents, tangents from outside the circle, tangents from outside

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.

Summary

Circle
A circle is a collection of all points in a plane which are at a constant distance(radius) from a fixed point (centre).

Secant
A line which intersects a circle in two distinct points is called secant of the circle.

Tangent
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.

Length of the tangent
The length of the segment of the tangent from the external point and the point of contact with the circle is called
the length of the tangent from the external point to the circle.

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.


radius perpendicular to tangent, radius and tangent, diameter and tangent   tangents from external point, external point, tangents to a circle, tangents, tangents of equal length, equality of two tangents, tangents from outside the circle, tangents from outside

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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