Use central, inscribed, and tangent angles

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Difficulty: Intermediate
Estimated time: 30 minutes
Domains: Geometry and Trigonometry
Techniques: Central-anglesInscribed-anglesTangent-radiusEqual-tangentsChord-bisectors

What you’ll learn

Tell whether an angle’s vertex is at the center of a circle or on the circle.
Match a central or inscribed angle to the arc it intercepts.
Use the right angle a radius makes with a tangent.
Use the fact that two tangent segments from one outside point are equal.
Use the right angle a line from the center makes with a chord that isn’t a diameter, at the chord’s midpoint.
Combine a circle fact with an angle sum you already know to find the angle or arc the question asks for.

Prerequisites

Why this matters on the SAT

Find the relationship before the number

Circle diagrams on the SAT can look crowded, with radii, chords, and lines everywhere. But one detail usually decides the whole solution, like where an angle’s vertex sits. Here’s a typical question.

Solution to the example

Start with the vertex. $C$ sits on the circle, not at the center, so $\angle ACB$ is an inscribed angle. An inscribed angle is half its arc:

m\angle ACB=\frac12(124^\circ)=62^\circ.

The answer is B. Choice C is what you’d get if the vertex were at the center, and choice D doubles the arc when it should be halved.

SAT example

The angle’s sides meet the circle at

A

and

B

, so the angle intercepts the thick arc from

A

B

Points $A$ , $B$ , and $C$ lie on a circle with center $O$ . Inscribed angle $\angle ACB$ intercepts minor arc $\widehat{AB}$ . If the measure of minor arc $\widehat{AB}$ is $124^\circ$ , what is $m\angle ACB$ ?

A
$54^\circ$
B
$62^\circ$
C
$124^\circ$
D
$248^\circ$

Spot the feature that decides the fact

First, a word about arcs. An arc measure tells you how much of the circle’s $360^\circ$ turn lies between two endpoints, so it’s in degrees. It isn’t the distance along the circle: a quarter of any circle measures $90^\circ$ , whether the circle is tiny or huge.

Any two points split a circle into two arcs that together make $360^\circ$ . Unless the points are the ends of a diameter, one is a shorter minor arc and the other a longer major arc. A major arc gets a third letter to show its path: $\widehat{ACB}$ runs from $A$ to $B$ through $C$ .

Every question here turns on one of five features, so look for it before you calculate:

An angle has its vertex at the center, like $\angle AOB$ in a circle with center $O$ .
An angle has its vertex on the circle, like $\angle ACB$ with $C$ on the circle.
A line touches the circle at exactly one point.
Two such lines come from the same point outside the circle.
A line from the center goes through the midpoint of a chord that isn’t a diameter.

If a question asks for the circumference or area of a whole circle, go back to Find perimeter and area of plane figures. Arc length, sector area, and radians come in the next lesson, and standard-form circle equations are in Read and write circle equations.

Check your understanding:

A problem gives a radius of $9$ and a central angle of $80^\circ$ , then asks for the physical length of the intercepted arc. Is the arc’s degree measure enough to answer it?

Central angles, inscribed angles, and intercepted arcs

An angle intercepts the arc between the two points where its sides meet the circle. To find that arc, trace each side of the angle out to the circle.

Where the vertex sits tells you how the angle and its arc compare.

A central angle has its vertex at the center. It has the same degree measure as its arc:

\boxed{m\angle AOB=m\widehat{AB}}.

An inscribed angle has its vertex on the circle, and its sides are chords, segments that join two points on the circle. It’s half its arc:

\boxed{m\angle ACB=\frac12m\widehat{AB}}.

So at the center, it’s the same; on the circle, it’s half. When a central angle and an inscribed angle intercept the same arc, the central angle is twice the inscribed one:

m\angle AOB=2m\angle ACB.

You can see this in the first panel below: an $80^\circ$ central angle and a $40^\circ$ inscribed angle share one arc.

Each panel pairs one feature with the fact it gives you.

Common mistake:

It’s tempting to halve every angle near a circle. Only an inscribed angle is half its arc. A central angle equals its arc. So check where the vertex is before you decide whether to halve.

Check your understanding:

A central angle and an inscribed angle intercept the same $96^\circ$ arc. What are their measures?

Right angles from tangents and chord midpoints

A tangent touches a circle at exactly one point, the point of tangency. Draw the radius to that point, and it meets the tangent at a right angle:

\boxed{\text{radius}\perp\text{tangent}}.

That right angle is often the key. It can build a right triangle, or give you two $90^\circ$ angles in a four-sided figure.

Now draw two tangent segments from the same outside point $P$ , touching the circle at $A$ and $B$ . They’re equal:

\boxed{PA=PB}.

So triangle $PAB$ is isosceles, and its base angles at $A$ and $B$ are equal.

A chord that isn’t a diameter gives you one more right angle. A line from the center through the chord’s midpoint is perpendicular to the chord. It works the other way too: a perpendicular from the center to a chord cuts the chord in half.

Why rule out a diameter? Its midpoint is the center itself, so every line through the center passes through that midpoint, and most of those lines aren’t perpendicular to the diameter.

The two equal halves of the chord make two mirror-image right triangles, so the central angle splits into two equal parts as well.

Common mistake:

Not every line that crosses the picture is a tangent. A tangent meets the circle at exactly one point. Once you’ve found one, draw the radius to that point and mark the $90^\circ$ angle.

Check your understanding:

Chord $\overline{JK}$ is not a diameter. A segment from center $O$ meets $\overline{JK}$ at its midpoint $M$ . What angle measure is guaranteed at $M$ ?

Example: Match two angles to one arc

Worked example

The two vertices differ, but the endpoints

A

and

B

show that both angles intercept the same arc.

In a circle with center $O$ , central angle $\angle AOB$ measures $(4x+12)^\circ$ , and inscribed angle $\angle ACB$ measures $(x+18)^\circ$ . Both angles intercept minor arc $\widehat{AB}$ . What is the measure of minor arc $\widehat{AB}$ , in degrees?

Step 1

Find each vertex

$O$ is the center, so $\angle AOB$ is a central angle. $C$ is on the circle, so $\angle ACB$ is inscribed.

Step 2

Check that they share an arc

Trace the sides of both angles out to the circle. They land on $A$ and $B$ both times, so both angles intercept minor arc $\widehat{AB}$ .

That means the central angle is twice the inscribed one:

4x+12=2(x+18).

Step 3

Solve for x

\begin{aligned} 4x+12&=2x+36\\[1.4em] 2x&=24\\[1.4em] x&=12. \end{aligned}

So the inscribed angle is $12+18=30^\circ$ , and the central angle is $4(12)+12=60^\circ$ . The doubling checks out: $60=2(30)$ .

Step 4

Answer with the arc

The question asks for the arc, so don’t stop at $x=12$ . A central angle has the same degree measure as its arc:

m\widehat{AB}=m\angle AOB=60^\circ.

Enter $\boxed{60}$ .

Check your understanding:

Now say an inscribed angle that intercepts minor arc $\widehat{AB}$ measures $37^\circ$ . What would the central angle and the minor arc measure?

Choose the theorem before the tool

These problems are quickest by hand, and they all run the same way:

Mark the angle or arc the question asks for.
Find the feature that decides the fact: the vertex, a point of tangency, or a chord’s midpoint.
Name the arc the angle intercepts, or the right angle you’re guaranteed.
Write one equation or angle sum.
Solve it, then go back to what the question asks for.

Why not a graphing calculator? It can’t tell you which arc an angle intercepts, or whether a line is a radius, a tangent, or a chord. That’s the real work, and once it’s done, the math is usually one doubling, halving, or angle sum. The calculators in the practice are optional, for checking your arithmetic.

Practice problems

For each problem, name the circle fact you’re using before you calculate.

Turn an arc into an angle

Practice problem

Points $K$ , $L$ , and $M$ lie on a circle. Inscribed angle $\angle KML$ intercepts minor arc $\widehat{KL}$ . If $m\widehat{KL}=146^\circ$ , what is $m\angle KML$ , in degrees?

Your answer

Calculator loads as you approach

This one is faster by hand. Use the calculator only if it helps you check the halving.

Get from equal tangents to an arc

Practice problem

Equal tangents create an isosceles triangle, while the two radii create right angles.

From point $P$ outside a circle with center $O$ , segments $\overline{PA}$ and $\overline{PB}$ are tangent to the circle at $A$ and $B$ , respectively. Segment $\overline{AB}$ is drawn. If $m\angle PAB=36^\circ$ , which choice gives the measure of minor arc $\widehat{AB}$ ?

Calculator loads as you approach

Find the equal angles and the right angles first. Check the arithmetic here if you want.

Build a major arc from a bisected chord

Practice problem

This chord isn’t a diameter. The radius through its midpoint splits the central angle into two equal halves.

In a circle with center $O$ , chord $\overline{AB}$ is not a diameter. Radius $\overline{OD}$ bisects $\overline{AB}$ at $M$ . Point $C$ lies on the major arc from $A$ to $B$ . If $m\angle AOD=38^\circ$ , what is the measure of major arc $\widehat{ACB}$ , in degrees?

Your answer

Calculator loads as you approach

The theorem does the work here. Use the calculator only to check the last step.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

A central angle equals its arc.
An inscribed angle is half its arc.
To find an angle’s arc, trace its sides out to the circle.
A radius to a point of tangency is perpendicular to the tangent.
Two tangent segments from the same outside point are equal.
For a chord that isn’t a diameter, a line from the center through its midpoint is perpendicular to it and splits the central angle into matching halves.
Arc measure is in degrees, not a length. Arc length, sector area, and radians come in the next lesson.
Find the circle fact first. The calculator is optional, only for checking the arithmetic that follows.

Next lesson

Find arc length, sector area, and radian measure

Use the fraction of a circle to connect angle measure with physical arc length, sector area, and radians.

Start next lesson

Practice

Practice this lesson

131 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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