01 · DEFINITION
What is a Tangent to a Circle?
A tangent to a circle is a straight line that intersects the circle at exactly one point.
Point of Contact
Point of Contact (P)
The single common point where the tangent line touches the circle.Secant vs Tangent
Secant vs Tangent
A secant intersects the circle in two points. A tangent is a special case of secant when the two end points of its corresponding chord coincide.02 · THEOREM 10.1
Theorem 10.1 & Complete Proof
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
THEOREM 10.1 DIAGRAM
Circle with Center O, Tangent XY touching at P
Given: Tangent XY touching circle C(O, r) at P.
To Prove: OP ⊥ XY
To Prove: OP ⊥ XY
Step 1
Take any point Q on tangent XY other than P. Connect OQ.
Step 2
Point Q must lie outside the circle (if Q lay inside, line XY would become a secant!).
Step 3
Therefore, OQ > radius OP for every point Q on XY except P.
Step 4
Since OP is the shortest distance from O to line XY, OP ⊥ XY (Proved).
03 · SOLVED EXAMPLE
Step-by-Step Problem Solving
EXAMPLE 1
A tangent PQ at point P of a circle of radius 5 cm meets a line through center O at point Q so that OQ = 12 cm. Find length PQ.
By Theorem 10.1, OP ⊥ PQ, so △OPQ is a right triangle at ∠P = 90°.
Using Pythagoras Theorem: OQ² = OP² + PQ²
12² = 5² + PQ²
144 = 25 + PQ²
PQ² = 119 ⟹ PQ = √119 cm (≈ 10.91 cm)
144 = 25 + PQ²
PQ² = 119 ⟹ PQ = √119 cm (≈ 10.91 cm)
04 · QUICK SUMMARY
Key Takeaways
01
A tangent touches a circle at exactly one point.
02
Radius through point of contact is perpendicular to tangent (OP ⊥ XY).
03
At any point on a circle, there is only one tangent.