Circles, Chords, and Tangents

Introduction

A circle is more than a curved shape. Its chords, radii, secants, and tangents create a network of angle and length relationships that you can prove and use. In this Grade 9–10 aiMOOC, you will learn how to recognize these relationships, explain why they are true, and apply them in unfamiliar geometry problems.

You should already be comfortable with angles, triangles, the Pythagorean theorem, basic triangle congruence, and simple algebra. The course also connects to coordinate geometry, trigonometry, and mathematical proof.

The diagram above combines a circle, a chord, and a tangent. As you work through the course, ask yourself what changes when a point moves and what remains invariant.

Learning Goals

By the end of this course, you should be able to define and distinguish a radius, diameter, chord, secant, tangent, and point of tangency. You should be able to use theorems about chords and tangents, justify those theorems with geometric reasoning, and solve multi-step problems involving lengths and angles.

You should also be able to communicate a proof clearly, make a precise construction with a compass and straightedge or dynamic geometry software, and explain how circle geometry appears in design, engineering, mapping, and other real situations.

Core Ideas

Circle Vocabulary

A circle is the set of all points in a plane that are the same distance from a fixed point called the center. That fixed distance is the radius.

A diameter is a chord that passes through the center. Its length is twice the radius, so d = 2r. Because it passes through the center, the diameter is the longest possible chord of a circle.

A chord is a line segment whose endpoints lie on the circle. A secant is a line that intersects a circle at two points. A tangent is a line that meets a circle at exactly one point, called the point of tangency.

Chords and Their Properties

Chords become especially useful when you compare their lengths with their distances from the center. In the same circle, equal chords are equally distant from the center. Conversely, chords that are equally distant from the center have equal lengths.

A central theorem says that a line from the center that is perpendicular to a chord bisects that chord. This creates two right triangles. If the circle has radius r, the perpendicular distance from the center to a chord is d, and half of the chord has length x, then the Pythagorean theorem gives x² + d² = r². Therefore the full chord length is 2x.

Chord Length and Central Angle

Longer chords subtend larger central angles, as long as you compare minor central angles in the same circle. A diameter subtends a straight angle of 180 degrees and is the longest chord.

If trigonometry is part of your course, a chord of length c in a circle of radius r with central angle θ satisfies c = 2r sin(θ/2). This formula comes from splitting the isosceles triangle formed by two radii and the chord into two right triangles.

Intersecting Chords

When two chords AB and CD intersect inside a circle at E, the products of the two segment lengths are equal:

AE × EB = CE × ED.

This is a form of the power of a point theorem. It lets you find a missing segment length even when no angle measures are given.

For example, if AE = 3, EB = 8, and CE = 4, then 3 × 8 = 4 × ED. Therefore ED = 6.

Tangents and Points of Tangency

A tangent line is perpendicular to the radius drawn to the point of tangency. If OT is a radius and a line touches the circle only at T, then OT forms a 90-degree angle with the tangent.

This theorem gives you an immediate right triangle whenever a tangent segment is connected to the center. That often allows you to use the Pythagorean theorem.

Tangents from an External Point

From a point P outside a circle, exactly two tangent segments can be drawn to the circle. If they touch the circle at A and B, then PA = PB.

One proof connects P to the center O. The radii OA and OB are equal, both are perpendicular to their tangents, and OP is shared. The two right triangles are congruent, so the tangent segments PA and PB are equal.

If two tangents PA and PB meet at P and the corresponding radii are OA and OB, then the quadrilateral OAPB contains two right angles. This means the angle between the tangents and the central angle satisfy ∠APB + ∠AOB = 180 degrees.

Angles with Tangents and Chords

The tangent-chord theorem, also called the alternate segment theorem in some curricula, connects a tangent angle to an inscribed angle. The angle between a tangent and a chord through the point of tangency equals the inscribed angle subtending that same chord in the opposite segment of the circle.

Equivalently, the measure of an angle formed by a tangent and a chord is half the measure of its intercepted arc. This result links tangent geometry to inscribed angles and arc measures.

Angle Between Two Tangents

When two tangents from the same external point touch a circle at A and B, their angle is supplementary to the central angle ∠AOB. Therefore, if ∠AOB = 124 degrees, then the angle between the tangents is 56 degrees.

This relationship is useful because a problem may give you an arc or central angle but ask for an exterior tangent angle.

Problem-Solving Strategies

First identify what kind of line or segment you are looking at. Ask whether it is a chord, radius, secant, or tangent. Then mark all information that follows automatically: equal radii, right angles at points of tangency, equal tangent segments from one external point, or bisected chords.

Next look for triangles. Circle theorems often create right triangles or isosceles triangles. Use congruence, the Pythagorean theorem, angle sums, or similarity only after you have justified why those relationships apply.

Finally, check whether your answer is geometrically reasonable. A chord cannot be longer than the diameter. A length must be positive. An angle inside a triangle must be less than 180 degrees, and a tangent-radius angle at the point of tangency must be exactly 90 degrees.

Worked Examples

Example: Distance from the Center to a Chord

A circle has radius 10 cm. A chord has length 16 cm. The perpendicular from the center bisects the chord, so half the chord is 8 cm. Let d be the distance from the center to the chord. Then 8² + d² = 10², so d² = 36 and d = 6 cm.

Example: Tangent Length

A point P lies 13 cm from the center O of a circle with radius 5 cm. PT is tangent at T. Because OT is perpendicular to PT, triangle OPT is right. Using the Pythagorean theorem, PT² + 5² = 13², so PT² = 144 and PT = 12 cm.

Example: Equal Tangents

From point P, tangents PA and PB touch the same circle. If PA = 7.5 cm, then PB = 7.5 cm. You do not need additional calculations because tangent segments from the same external point are congruent.

Example: Tangent-Chord Angle

A tangent and a chord form a 38-degree angle at the point of tangency. Any inscribed angle subtending the same chord in the opposite segment also measures 38 degrees. The intercepted arc has measure 76 degrees because the tangent-chord angle is half its intercepted arc.

Example: Tangent-Secant Power

A tangent PT and a secant through points A and B start from the same external point P. If PA = 4 cm and PB = 9 cm, then PT² = PA × PB = 36, so PT = 6 cm. Here PB is the whole secant length from P to the farther intersection point.

Common Misconceptions

A tangent is not simply a line that looks as if it barely touches a circle. Its defining geometric relationship is that it meets the circle at one point and is perpendicular to the radius at that point.

A chord is not always a diameter. Every diameter is a chord, but only the chord through the center is a diameter.

The equal-tangent theorem applies only to tangent segments drawn from the same external point to the same circle. Do not assume unrelated tangent segments are equal.

When using a tangent-secant product such as PT² = PA × PB, PA is the external part of the secant and PB is the entire secant from the external point to the farther intersection.

Interactive Tasks

Quiz: Test Your Knowledge

Which statement correctly defines a chord of a circle? (A segment with both endpoints on the circle) (!A line that touches the circle at one point) (!A segment from the center to the circle) (!A line that never meets the circle)

Which chord is always the longest chord in a circle? (The diameter) (!The radius) (!The tangent) (!The secant)

What angle is formed by a radius and a tangent at the point of tangency? (90 degrees) (!45 degrees) (!120 degrees) (!180 degrees)

What does a perpendicular from the center of a circle do to a chord? (It bisects the chord) (!It doubles the chord) (!It makes the chord a tangent) (!It makes the chord a radius)

Two tangent segments are drawn from the same external point to one circle. How are their lengths related? (They are equal) (!One is twice the other) (!Their sum equals the radius) (!Their product equals the diameter)

A circle has radius 7 cm. What is its diameter? (14 cm) (!7 cm) (!21 cm) (!49 cm)

A tangent and a chord form a 35-degree angle. What is the measure of the intercepted arc? (70 degrees) (!35 degrees) (!105 degrees) (!145 degrees)

Two tangents from an external point form a 58-degree angle. What is the central angle between the radii to the tangent points? (122 degrees) (!58 degrees) (!90 degrees) (!238 degrees)

Intersecting chords satisfy AE times EB equals CE times ED. If AE is 3, EB is 8, and CE is 4, what is ED? (6) (!4) (!8) (!12)

A point is 13 cm from the center of a circle of radius 5 cm. What is the tangent length from the point to the circle? (12 cm) (!8 cm) (!13 cm) (!18 cm)

Memory Game

Chord	Segment with both endpoints on the circle
Tangent	Line that meets a circle at one point
Secant	Line that intersects a circle at two points
Diameter	Chord that passes through the center
Point of tangency	Point where a tangent touches the circle
Perpendicular bisector	Line that crosses a segment at its midpoint at a right angle
Power of a point	Relationship connecting products of secant, chord, or tangent lengths

Drag and Drop

Match the correct terms.	Topic
Radius to point of tangency	Perpendicular to the tangent
Equal chords in one circle	Equidistant from the center
Two tangents from one external point	Equal tangent lengths
Tangent and chord angle	Half the intercepted arc measure
Intersecting chord segments	Equal products of segment lengths

Crossword Puzzle

Tangent	Which line meets a circle at exactly one point?
Chord	Which segment has both endpoints on a circle?
Radius	Which segment joins the center to a point on the circle?
Diameter	Which chord passes through the center?
Secant	Which line intersects a circle at two points?
Perpendicular	What relationship describes a radius and tangent at the point of tangency?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Circle: Create a labeled one-page diagram showing a center, radius, diameter, chord, secant, tangent, arc, and point of tangency, then add one sentence explaining each label.
Geometry in everyday life: Photograph or sketch four real circular objects and identify where a chord, secant, or tangent could be represented on each object.
Mathematical communication: Record a 60–90 second explainer video showing why a diameter is a special kind of chord and why every radius in one circle has the same length.
Measurement: Draw three chords in the same circle, measure their lengths and distances from the center, and describe the pattern you observe.

Standard

Perpendicular bisector: Use a compass and straightedge or dynamic geometry software to construct the perpendicular bisector of a chord and document how the construction locates the center line.
Tangent lines to circles: Construct a tangent at a chosen point on a circle, explain why it is perpendicular to the radius, and include a labeled image of your result.
Inscribed angle: Create a short illustrated proof or slide explaining the tangent-chord theorem and test it with at least three different chord positions.
Interview: Interview a designer, engineer, craftsperson, technician, or teacher about where circular curves and tangency appear in their work, then connect one example to a theorem from this course.

Advanced

Power of a point: Design and solve a set of three connected problems that use intersecting chords, a tangent-secant relationship, and a written explanation of why the product equations work.
Coordinate geometry: Place a circle on a coordinate plane, choose a point on it, derive or verify the equation of the tangent line at that point, and explain how perpendicular slopes support your result.
Dynamic geometry: Build an interactive model in which a tangent point or chord endpoint moves, then collect evidence for two invariants and write a conjecture followed by a proof.
Mathematical modelling: Visit or study a local circular feature such as a roundabout, wheel, arch, track, or curved sign, create a scaled model, and explain how chords or tangents can estimate a useful distance or direction.

Learning Assessment

Proof with tangents: Given two tangents from one external point, construct a complete proof that their lengths are equal and identify every theorem used.
Chord distance problem: Solve a problem in which the radius and chord length are known but the distance from the center to the chord is unknown, then explain why the chord must be bisected before using the Pythagorean theorem.
Tangent-chord transfer: Analyze a diagram containing a tangent, chord, arc, and inscribed angle, determine two unknown angles, and justify each step rather than giving only numerical answers.
Error analysis: Critique a worked solution that incorrectly treats a secant as a tangent, identify the first invalid step, and replace it with a correct strategy.
Geometry design: Create a small design problem involving a circular path and a straight line that must be tangent, solve it, and explain how the 90-degree radius-tangent relationship guarantees tangency.
Theorem comparison: Compare the intersecting-chords theorem and the tangent-secant theorem by explaining what is structurally similar, what is different, and when each can be applied.

Evidence of Learning

Knowledge: You accurately use the vocabulary of circle geometry and state the main chord and tangent theorems with their conditions.
Skills: You construct diagrams, mark implied relationships, calculate unknown lengths and angles, and produce logically ordered proofs.
Products: You create labeled diagrams, a short explanation or video, a construction, and at least one multi-step problem solution that can be checked by another learner.
Transfer: You recognize when circle theorems apply in unfamiliar diagrams or real situations and justify why your chosen theorem is valid.

OERs on the Topic

The following English Wikipedia article provides an open reference for definitions, formulas, and further links about circles. Use it to review terminology and then compare its explanations with the proofs and examples in this course.

Linked Learning Areas

Circle geometry

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