Tangents to a Circle

Try this

Drag $B$ toward $A$ . As chord $A B$ gets shorter, what angle does $∠ O A B$ get closer and closer to?
Move $B$ right onto $A$ . How many common points do the line and the circle have now? What new mark appears in the figure?
Keep $B$ stuck to $A$ and drag $A$ all the way round the circle. The tangent turns with it. Does its angle with radius $O A$ ever change?

Why $9 0^{\circ}$ ? Since $O A = O B$ , triangle $△ O A B$ is isosceles, so

∠ O A B = \frac{18 0 ^{\circ} - ∠ A O B}{2} = 9 0^{\circ} - \frac{1}{2} ∠ A O B .

As $B$ approaches $A$ , $∠ A O B$ shrinks toward $0^{\circ}$ , so $∠ O A B$ climbs toward $9 0^{\circ}$ . When the points coincide, the secant has become the tangent and the angle is exactly a right angle:

Property: a tangent to a circle is perpendicular to the radius at the point of tangency.
Test: a line through the outer end of a radius and perpendicular to that radius is a tangent to the circle.

Break it

Counterexample lab: remove one condition

Conclusion: This line through $A$ has only one common point with the circle

The line passes through $A$ and is perpendicular to $O A$ : it is the tangent, and $A$ is its only common point.

Two tangents from a point outside

Drag

P

. Are the two tangent lengths ever different?

$△ O A P$ and $△ O B P$ are both right triangles with the common hypotenuse $O P$ , and $O A = O B$ , so they are congruent. Therefore

$P A = P B$ (the tangent length theorem);
$P O$ bisects $∠ A P B$ , and it also bisects $∠ A O B$ .

Predict first

$P A$ and $P B$ are tangents to circle $O$ at $A$ and $B$ , and $∠ A P B = 5 0^{\circ}$ . What is $∠ A O B$ , in degrees?

Choose an answer first; then you can see the result.

The tangent–chord angle

An angle with its vertex on the circle, one side a tangent and the other side a chord, is a tangent–chord angle. It is the “limiting case” of an inscribed angle: turn one side of an inscribed angle about its vertex until it touches the circle.

Proof: a tangent–chord angle equals the inscribed angle on the arc it encloses

Step 1 / 5

What to prove
$A T$ is a tangent, $A B$ is a chord, and $C$ is on the other arc. We want to prove the red tangent–chord angle equals the blue inscribed angle.

$∠ B A T = ∠ A C B ?$
Draw diameter AD and join DB
First right angle: tangent ⊥ radius
Second right angle: the angle on a diameter
Swap D for C

Practice

Exercise 1Basic

Circle $O$ has radius $5$ . Point $P$ is at distance $13$ from the center, and $P A$ is tangent to the circle at $A$ . Find the tangent length $P A$ .

Exercise 2Basic

$P A$ and $P B$ are tangent to circle $O$ at $A$ and $B$ , and $∠ A P B = 7 0^{\circ}$ . Find $∠ A O B$ in degrees.

Exercise 3Advanced

Line $A T$ is tangent to a circle at $A$ , $A B$ is a chord, and the tangent–chord angle $∠ B A T = 6 5^{\circ}$ . Point $C$ lies on the arc enclosed by $∠ B A T$ (and is not $A$ or $B$ ). Find $∠ A C B$ in degrees.

Exercise 4Challenge

In $△ A B C$ , $B C = 13$ , $C A = 14$ and $A B = 15$ . Its incircle touches $A B$ at $D$ . Find the length of $A D$ .

Try this

Counterexample lab: remove one condition

Two tangents from a point outside

The tangent–chord angle

Proof: a tangent–chord angle equals the inscribed angle on the arc it encloses

Practice

Mistake Museum · Exhibits from this stop

Proving a tangent with half the condition missing

Matching the tangent–chord angle with the wrong arc