Circle Angles, Chords and Tangents, SAT math lesson

A bicycle wheel touches the road at a single point, and the spoke to that point stands straight up from the road. A pizza cut through the center splits into two equal halves, and any cut that misses the center leaves a smaller piece. Pictures like these hide a handful of exact facts about circles. Most circle questions on the SAT come down to picking the one fact that links what is given to what is asked, and very often that fact produces a right triangle.

Key terms

Radius and diameter

Definition A circle is the set of all points in a plane at a fixed distance $r > 0$ from a fixed point, its center. A radius is a segment from the center to a point of the circle, and $r$ is also called the radius. A diameter is a segment through the center with both endpoints on the circle. Its length is

d = 2r.

All radii of one circle have the same length, which is the fact behind many circle problems.

In plain words Every point on the circle is exactly the same distance from the center. A diameter is two radii laid end to end in a straight line.

Example A circle has diameter $18$ , so its radius is $9$ . If $O$ is the center and $A$ and $B$ are points on the circle, then $OA = OB = 9$ , so triangle $OAB$ is isosceles. If angle $AOB$ measures $40^\circ$ , the two base angles are each

\frac{180^\circ - 40^\circ}{2} = 70^\circ.

Central angle and intercepted arc

Definition A central angle is an angle whose vertex is the center of the circle. The arc an angle intercepts is the arc that lies in the interior of the angle, with its endpoints on the two sides of the angle. Two points $A$ and $C$ on a circle split it into two arcs. The minor arc is the shorter one, named with two letters as arc $AC$ , and the major arc is the longer one, named with three letters such as arc $ABC$ , where $B$ is a point on it. The degree measure of a minor arc equals the measure of its central angle, the major arc measures $360^\circ$ minus that, and a semicircle measures $180^\circ$ .

In plain words The angle at the center opens up like a slice of pizza, and the crust of that slice is the intercepted arc. Since two points always cut the circle into two arcs, say which one you mean.

Example Center $O$ and points $A$ and $C$ have angle $AOC = 80^\circ$ . Minor arc $AC$ measures $80^\circ$ and major arc $ABC$ measures $360^\circ - 80^\circ = 280^\circ$ . Arc measure in degrees is not arc length. On a circle of radius $9$ the minor arc has length

\frac{80}{360} \cdot 2\pi \cdot 9 = 4\pi.

Inscribed angle

Definition An inscribed angle is an angle whose vertex lies on the circle and whose sides contain chords of the circle. The inscribed angle theorem says that the measure of an inscribed angle is half the measure of its intercepted arc,

m\angle ABC = \frac{1}{2} \, m\overset{\frown}{AC},

where arc $AC$ is the arc inside the angle, the one that does not contain $B$ . Two consequences follow. Inscribed angles that intercept the same arc are equal. An inscribed angle that intercepts a semicircle, so that its sides end at the two ends of a diameter, is a right angle.

In plain words An angle at the edge of the circle sees the same arc as the angle at the center, but it is only half as big. Slide the vertex along the circle and the angle does not change, as long as it stays on the same side and looks at the same arc.

Example Points $A$ , $B$ and $C$ lie on a circle with center $O$ , and central angle $AOC$ measures $110^\circ$ . If $B$ is on the major arc, angle $ABC$ intercepts minor arc $AC$ and measures

\frac{110^\circ}{2} = 55^\circ.

If $B$ is on the minor arc instead, angle $ABC$ intercepts the major arc of $250^\circ$ and measures $125^\circ$ . The two answers add to $180^\circ$ , which is why naming the intercepted arc matters.

Chord

Definition A chord is a segment whose endpoints both lie on the circle. A diameter is the longest chord. The segment from the center perpendicular to a chord bisects the chord, and its length is the distance from the center to the chord. If a chord of a circle with radius $r$ is at distance $h$ from the center, where $0 \le h < r$ , then half the chord, $h$ and a radius form a right triangle with hypotenuse $r$ , so the chord has length

2\sqrt{r^2 - h^2}.

In plain words Draw the radius to one end of the chord and the perpendicular from the center to the chord. Together with half the chord they make a right triangle, and the radius is its hypotenuse.

Example A circle has radius $10$ and a chord $6$ units from the center. Half the chord is

\sqrt{10^2 - 6^2} = \sqrt{64} = 8,

so the chord has length $16$ . Forgetting to double gives $8$ , a common wrong choice.

Tangent line

Definition A line is tangent to a circle when it lies in the plane of the circle and meets the circle in exactly one point, the point of tangency. A tangent line is perpendicular to the radius drawn to the point of tangency. Conversely, a line through a point of the circle that is perpendicular to the radius at that point is tangent to the circle. From a point outside the circle there are exactly two tangent lines, and the two tangent segments from that point to the points of tangency are equal in length.

In plain words A tangent just touches the circle, and it always meets the radius at a right angle. So a tangent, a radius and the segment to the center form a right triangle with its right angle at the point of tangency.

Example Line $PT$ is tangent to a circle with center $O$ at $T$ , the radius is $5$ and $PO = 13$ . Angle $OTP$ is a right angle and $PO$ is the hypotenuse, so

PT = \sqrt{13^2 - 5^2} = 12.

Using only what is stated

Definition Each circle fact above has a condition. A point must lie on the circle, a segment must pass through the center, or a line must be tangent, and these incidences have to be stated in the question or marked in the figure. On the SAT a figure is drawn to scale unless the question says it is not, but a line that looks tangent or a segment that looks like a diameter is still not enough for an exact conclusion. When a figure is labeled not drawn to scale, its proportions may have been changed on purpose, so read no lengths or angles from it at all.

In plain words Read the words first. Use a right angle at a point of tangency only when the line is said to be tangent, and use a right inscribed angle only when the segment is said to be a diameter.

Example Points $A$ , $B$ and $C$ lie on a circle, and $AC$ is a diameter. Then angle $ABC$ is $90^\circ$ wherever $B$ sits on the circle. If $AB = 6$ and $BC = 8$ , the Pythagorean theorem gives $AC = 10$ , so the radius is $5$ . Without the word diameter, the right angle at $B$ would be a guess.

Check yourself

Points $A$ , $B$ and $C$ lie on a circle with center $O$ . Central angle $AOC$ measures $124^\circ$ , and $B$ lies on the major arc $ABC$ . What is the measure of angle $ABC$ ? (A) $56^\circ$ (B) $62^\circ$ (C) $118^\circ$ (D) $124^\circ$

Show the answer

The correct choice is (B). With $B$ on the major arc, angle $ABC$ intercepts minor arc $AC$ , which measures $124^\circ$ , and an inscribed angle is half its arc. Choice (D) treats the inscribed angle as a central angle. Choice (C) is the angle for a vertex on the minor arc, half of $236^\circ$ . Choice (A) is $180^\circ - 124^\circ$ , which has no meaning here.

Line $PT$ is tangent at $T$ to a circle with center $O$ . The radius $OT$ is $8$ and $PT = 15$ . What is the length of $PO$ ?

Show the answer

The answer is $17$ . The radius to the point of tangency is perpendicular to the tangent, so triangle $OTP$ has its right angle at $T$ and hypotenuse $PO$ . Then

PO = \sqrt{8^2 + 15^2} = \sqrt{289} = 17.

A chord of length $24$ lies in a circle of radius $13$ . What is the distance from the center of the circle to the chord? (A) $1$ (B) $5$ (C) $11$ (D) $12$

Show the answer

The correct choice is (B). The perpendicular from the center bisects the chord, so half the chord is $12$ and the radius $13$ is the hypotenuse. The distance is $\sqrt{13^2 - 12^2} = \sqrt{25} = 5$ . Choice (D) is half the chord, choice (A) subtracts the lengths instead of their squares, and choice (C) is $24 - 13$ .

Circle Angles, Chords and Tangents

Theory, part 103

Practice, part 104

Key terms

Radius and diameter

Central angle and intercepted arc

Inscribed angle

Chord

Tangent line

Using only what is stated

Check yourself

How this lesson connects

Builds on

Used later in