Unit 10 is about circles, from their equations on the coordinate plane to the angles, arcs, and segments that show up inside and around them. By the end, you should be able to write and read circle equations, find any angle or arc using the right formula, and solve for missing lengths with tangents, secants, and chords.
Read the overview for the big picture, then work through each topic's notes and test yourself with the checkpoint questions before checking the common mistakes.
This unit pulls together algebra and classic geometry. You'll use the distance formula to describe circles on a grid, then use theorems about arcs, angles, and segments to solve multi-step problems and write proofs.
Honors Geometry Unit 10 covers circles. You learn the standard form and how to complete the square to find a circle's center and radius. You learn how central, inscribed, and other angles relate to their intercepted arcs depending on where the vertex sits. You study tangent and secant lines, including the tangent-radius right angle and the segment product theorems. You also work with the parts of a circle, chord properties, arc length, and sector area.
Any point on the circle is exactly units from the center . Squaring the distance formula gives . When the equation is expanded, completing the square puts it back in this form so you can read off the center and radius.
A central angle has the same degree measure as its intercepted arc. An inscribed angle or tangent-chord angle measures half its intercepted arc. For two chords crossing inside a circle, use half the sum of the arcs intercepted by the angle and its vertical angle. For secants or tangents meeting outside, use half the difference of the far and near arcs.
A tangent is perpendicular to the radius at the point of tangency, which sets up the Pythagorean theorem. When chords, secants, or tangents meet, multiplying the right segment lengths gives equal products you can solve for an unknown.
Because every radius has the same length, you get the circle equation, isosceles triangles inside circles, and the inscribed angle theorem. When a circle problem feels stuck, draw a radius to a point you care about and see what congruent segments or right angles appear.
A circle's equation is the distance formula in disguise.
Circle-angle formulas depend on the location of the vertex and how the angle's sides meet the circle.
Tangents and secants give you two kinds of facts, right angles and segment products.
A circle is every point in a plane at a fixed distance, the radius, from one center point.
Open this guide for a closer review of the topic.
A circle's equation is the distance formula in disguise. Every point on a circle with center and radius sits exactly units from the center, so the Pythagorean theorem gives the standard form .
Reading the center takes care with signs. The equation subtracts and , so means . The right side is , not , so take a square root to get the radius.
If the equation is expanded, like , complete the square to get back to standard form.
Worked example. Find the center and radius of . Group the terms and move the constant to get . Half of 8 is 4, and . Half of is , and . Add 16 and 1 to both sides to get . The center is and the radius is 5.
To test whether a point is on the circle, plug it in. The point gives , so it lies on the circle.
Check yourself. What are the center and radius of ? (Answer: The center is and the radius is 7.)
Circle-angle formulas depend on the location of the vertex and how the angle's sides meet the circle. Find the vertex and identify the configuration before picking a formula.
A central angle has its vertex at the center and equals its intercepted arc. An inscribed angle has its vertex on the circle and sides that pass through the circle. It measures half its intercepted arc, so a 100° arc gives a 50° inscribed angle. A tangent-chord angle also has its vertex on the circle and follows the same half-the-arc rule.
Three corollaries come up constantly. Inscribed angles that intercept the same arc are congruent. An inscribed angle that intercepts a semicircle is a right angle. Opposite angles of a quadrilateral inscribed in a circle are supplementary.
When the vertex is inside the circle, as with two crossing chords, the angle is half the sum of the arc it intercepts and the arc its vertical angle intercepts. When the vertex is outside, with two secants, two tangents, or one of each, the angle is half the difference of the far arc and the near arc.
Worked example. Two chords cross inside a circle. The angle intercepts a 100° arc and its vertical angle intercepts a 40° arc. The angle is . Now suppose two secants meet outside a circle, cutting off a far arc of 130° and a near arc of 50°. That angle is .
Check yourself. Quadrilateral ABCD is inscribed in a circle and angle A measures 72°. What is the measure of angle C? (Answer: 108°, because opposite angles of an inscribed quadrilateral are supplementary.)
| Vertex location | Angles that fit | Angle measure |
|---|---|---|
| At the center | Central angle | Equal to the intercepted arc |
| On the circle | Inscribed angle, tangent-chord angle | Half the intercepted arc |
| Inside the circle | Two chords crossing | Half the sum of the two intercepted arcs |
| Outside the circle | Two secants, two tangents, or a secant and a tangent | Half the difference of the far arc and the near arc |
Tangents and secants give you two kinds of facts, right angles and segment products.
A tangent touches the circle at exactly one point and is perpendicular to the radius at that point. That right angle lets you use the Pythagorean theorem. It also lets you write a tangent line's equation on the coordinate plane. When the radius has a nonzero slope, the tangent's slope is its negative reciprocal. A horizontal radius has a vertical tangent, and a vertical radius has a horizontal tangent. Two tangent segments drawn from the same outside point are congruent.
The segment theorems all multiply lengths. When two chords cross inside a circle, the two pieces of one chord multiply to the same product as the two pieces of the other. For segments from an outside point, multiply each whole segment by its outside part. A tangent counts as both its own whole and its own outside part, so it gets squared.
Worked example. From point , a secant has an outside part of 4 and an inside part of 5, and a tangent from touches the circle at . The whole secant is , so and . A common slip is writing , but the inside part alone never goes into this equation.
For a radius-and-tangent problem, if the radius is 5 and the center is 13 units from , the tangent segment from is .
Check yourself. Two chords intersect inside a circle. One chord is split into pieces of 3 and 8, and the other into pieces of 4 and . What is ? (Answer: , so .)
| Setup | Where the segments meet | Equation |
|---|---|---|
| Two chords | Inside the circle | part × part = part × part |
| Two secants | Outside the circle | whole × outside = whole × outside |
| Tangent and secant | Outside the circle | tangent² = whole × outside |
| Two tangents | Outside the circle | The two tangent segments are equal |
A circle is every point in a plane at a fixed distance, the radius, from one center point. Radii connect the center to the circle, and chords connect two points on the circle. A diameter is a chord through the center, so and it's the longest chord.
Arcs come in three sizes. A minor arc is less than 180°, a semicircle is exactly 180°, and a major arc is more than 180° and gets named with three letters, like arc . Arc measure is in degrees and matches the corresponding central angle. For a major arc, use the reflex central angle, which measures more than 180°. Arc length is a distance, a fraction of the circumference.
Chords have their own rules. A radius or diameter perpendicular to a chord bisects that chord. Congruent chords sit the same distance from the center and cut off congruent arcs.
Worked example. In a circle with radius 6, a central angle measures 120°. The arc measure is 120°. The arc length is . The sector area is .
A polygon is inscribed in a circle when all its vertices lie on the circle. A polygon is circumscribed about a circle when every side is tangent to the circle.
Check yourself. A circle has radius 10. What is the area of a sector with a 72° central angle? (Answer: square units.)
| Term | Definition |
|---|---|
| Radius | A segment from the center of a circle to any point on the circle, or the length of that segment. |
| Diameter | A chord that passes through the center, equal to twice the radius. |
| Central angle | An angle with its vertex at the center of a circle whose measure equals its intercepted arc. |
| Inscribed angle | An angle with its vertex on a circle and sides that are chords, measuring half its intercepted arc. |
| Point of tangency | The single point where a tangent line touches a circle. |
| Minor arc | An arc measuring less than 180°, named with its two endpoints. |
| Major arc | An arc measuring more than 180°, named with three points to show which way around the circle it goes. |
| Circumscribed angle | An angle formed by two tangents from the same outside point, equal to 180° minus the minor arc it intercepts. |
| Concentric circles | Circles in the same plane that share a center but have different radii. |
Standard form subtracts the center coordinates, so has center , not . Rewrite each plus as minus a negative, like , before reading off and .
The number on the right side of standard form is the radius squared. If it's 25, the radius is 5. The same idea applies after completing the square, where you also have to add the new constants to both sides or the radius comes out wrong.
For segments from an outside point, the product is whole × outside, where the whole runs from the outside point all the way to the far side of the circle. Add the outside and inside parts first, then multiply. The part × part rule only applies to chords crossing inside the circle.
Both formulas use two arcs, so they're easy to swap. Vertex inside the circle means half the sum, and vertex outside means half the difference. Locate the vertex before you write anything down.