Tangent to a Circle Formula

Tangent to a circle is a line that touches a circle at exactly one point, called the point of tangency.

The Formula

tangent⊥radius at point of tangency\text{tangent} \perp \text{radius at point of tangency}

When to use: Imagine a ball sitting on a flat floor. The floor touches the ball at exactly one point—that's tangency. The floor (tangent line) is perfectly perpendicular to a line from the ball's center to the contact point (the radius). No matter how you tilt the flat surface, if it only touches at one point, it must be perpendicular to the radius there.

Quick Example

A circle centered at the origin with radius 55. The line x=5x = 5 is tangent at (5,0)(5, 0): radius to (5,0) is horizontal⊥tangent line x=5 (vertical)\text{radius to } (5,0) \text{ is horizontal} \perp \text{tangent line } x = 5 \text{ (vertical)}

Notation

Tangent line at point PP is denoted ℓP\ell_P; the key property is ℓP⊥OP\ell_P \perp OP where OO is the center

What This Formula Means

A line that touches a circle at exactly one point, called the point of tangency. At this point, the tangent line is perpendicular to the radius.

Imagine a ball sitting on a flat floor. The floor touches the ball at exactly one point—that's tangency. The floor (tangent line) is perfectly perpendicular to a line from the ball's center to the contact point (the radius). No matter how you tilt the flat surface, if it only touches at one point, it must be perpendicular to the radius there.

Formal View

Line ℓ\ell is tangent to circle S1(O,r)S^1(O,r) at PP iff ℓ∩S1={P}\ell \cap S^1 = \{P\} and OP→⊥ℓ\overrightarrow{OP} \perp \ell; from external point EE: two tangent segments ∣ET1∣=∣ET2∣=∣OE∣2−r2|ET_1| = |ET_2| = \sqrt{|OE|^2 - r^2}

Worked Examples

Example 1

easy
A tangent line touches circle OO at point PP. The radius OP=7OP = 7 cm. A line from an external point AA is tangent to the circle at PP, and OA=25OA = 25 cm. Find the length of the tangent segment APAP.

Answer

AP=24AP = 24 cm

First step

1
Step 1: Recall that a tangent to a circle is perpendicular to the radius at the point of tangency. So OP⊥APOP \perp AP, making triangle OAPOAP a right triangle with the right angle at PP.

Full solution

  1. 2
    Step 2: Identify the hypotenuse: OA=25OA = 25 cm (from center to external point), and one leg OP=7OP = 7 cm (radius).
  2. 3
    Step 3: Apply the Pythagorean theorem: AP2+OP2=OA2AP^2 + OP^2 = OA^2, so AP2=252−72=625−49=576AP^2 = 25^2 - 7^2 = 625 - 49 = 576.
  3. 4
    Step 4: Take the square root: AP=576=24AP = \sqrt{576} = 24 cm.
Because a tangent is perpendicular to the radius at the point of tangency, triangle OAP is right-angled at P. This is a classic 7-24-25 Pythagorean triple, giving AP = 24 cm.

Example 2

medium
Two tangent segments PAPA and PBPB are drawn from external point PP to circle OO. If PA=3x−4PA = 3x - 4 and PB=x+8PB = x + 8, find the lengths of both tangent segments.

Example 3

medium
A circle has center O(0,0)O(0,0) and radius 55. Verify that the line x=5x = 5 is tangent to the circle and identify the point of tangency.

Common Mistakes

  • Forgetting the right angle — at the point of tangency the tangent is always perpendicular to the radius, which is the key that builds the right triangle.
  • Calling a two-point line a tangent — a tangent touches at exactly one point; two intersection points make it a secant.
  • Drawing the perpendicular to the wrong segment — the right angle is between the tangent and the radius to the point of tangency, not to the diameter or another radius.

Why This Formula Matters

The tangent-perpendicular-to-radius fact converts a 'touching' picture into a right triangle, unlocking the Pythagorean theorem for tangent-length and distance problems; without the perpendicular insight, these circle problems have no foothold. Recognizing it by "Does the line meet the circle at exactly one point, making it perpendicular to the radius there?" — rather than by familiar numbers — is what lets a student tell it apart from chord / secant and inscribed angle and perpendicularity (general) in a mixed problem set.

Frequently Asked Questions

What is the Tangent to a Circle formula?

A line that touches a circle at exactly one point, called the point of tangency. At this point, the tangent line is perpendicular to the radius.

How do you use the Tangent to a Circle formula?

Imagine a ball sitting on a flat floor. The floor touches the ball at exactly one point—that's tangency. The floor (tangent line) is perfectly perpendicular to a line from the ball's center to the contact point (the radius). No matter how you tilt the flat surface, if it only touches at one point, it must be perpendicular to the radius there.

What do the symbols mean in the Tangent to a Circle formula?

Tangent line at point PP is denoted ℓP\ell_P; the key property is ℓP⊥OP\ell_P \perp OP where OO is the center

Why is the Tangent to a Circle formula important in Math?

The tangent-perpendicular-to-radius fact converts a 'touching' picture into a right triangle, unlocking the Pythagorean theorem for tangent-length and distance problems; without the perpendicular insight, these circle problems have no foothold. Recognizing it by "Does the line meet the circle at exactly one point, making it perpendicular to the radius there?" — rather than by familiar numbers — is what lets a student tell it apart from chord / secant and inscribed angle and perpendicularity (general) in a mixed problem set.

What do students get wrong about Tangent to a Circle?

The procedure for tangent to a circle is the easy part; the trap is forgetting the right angle. Asking "Does the line meet the circle at exactly one point, making it perpendicular to the radius there?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Tangent to a Circle formula?

Before studying the Tangent to a Circle formula, you should understand: circles, perpendicularity.

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