Circle theorems describe how angles, arcs, chords, tangents, and secants are related in and around a circle. Students need this reference because many geometry problems combine several circle rules in one diagram. A clear cheat sheet helps students identify which theorem matches the given information and choose the correct formula quickly.
The most important ideas are that central angles match their intercepted arcs, inscribed angles measure half their intercepted arcs, and tangent lines meet radii at right angles. Secant and tangent relationships use products of segment lengths, especially in power of a point problems. When solving, label arcs and segments carefully before substituting into formulas.
Key Facts
- A central angle has the same measure as its intercepted arc, so .
- An inscribed angle measures half its intercepted arc, so .
- An angle formed by two chords inside a circle equals half the sum of the intercepted arcs, so .
- An exterior angle formed by two secants, two tangents, or a tangent and a secant equals half the difference of the intercepted arcs, so .
- A tangent line is perpendicular to the radius at the point of tangency, so if is a radius and is tangent, then .
- For two intersecting chords inside a circle, the products of the chord segments are equal, so .
- For two secants from the same external point, the outside segment times the whole secant is equal for both secants, so .
- For a tangent and a secant from the same external point, the tangent squared equals the outside secant segment times the whole secant, so .
Vocabulary
- Central angle
- An angle whose vertex is at the center of a circle and whose sides are radii.
- Inscribed angle
- An angle whose vertex lies on the circle and whose sides are chords of the circle.
- Tangent
- A line that touches a circle at exactly one point, called the point of tangency.
- Secant
- A line that intersects a circle at exactly two points.
- Chord
- A segment whose endpoints both lie on the circle.
- Power of a point
- A relationship showing that products of certain tangent, secant, or chord segment lengths from the same point are equal.
Common Mistakes to Avoid
- Using the full arc measure for an inscribed angle is wrong because an inscribed angle is half its intercepted arc, so use .
- Adding arcs for an exterior angle is wrong because exterior circle angles use half the difference of the intercepted arcs, so use .
- Using only the outside secant segment in a secant product is wrong because the formula needs outside times whole, so use , not just unless is the whole secant.
- Forgetting that a tangent is perpendicular to the radius is wrong because the radius to the point of tangency forms a right angle, so .
- Mixing up chord and secant formulas is wrong because intersecting chords inside the circle use , while external secants use outside times whole.
Practice Questions
- 1 An inscribed angle intercepts an arc measuring . What is the measure of the inscribed angle?
- 2 Two chords intersect inside a circle. If the segments of one chord are and , and one segment of the other chord is , what is the missing segment length?
- 3 From point , a tangent has length and a secant has outside segment length . What is the whole secant length?
- 4 A problem gives a tangent, a secant, and a radius drawn to the point of tangency. Which circle theorem should you use first, and why?
Understanding Circle Theorems Reference
Most circle theorems come from one important feature of a circle. Every radius has the same length. This creates isosceles triangles whenever two radii join points on the circle.
The equal base angles in those triangles help explain many angle rules. A full turn around the centre contains three hundred sixty degrees, so the arcs of a circle must add to three hundred sixty degrees as well.
This total is useful when a diagram gives several arcs but leaves one unknown. A semicircle is especially important because its arc measures one hundred eighty degrees.
The location of an angle tells you which parts of the circle control it. An angle with its vertex on the circle looks across the circle to the arc opposite the vertex. Students often choose the small nearby arc by mistake.
Trace each side of the angle until it reaches the circle, then identify the arc lying inside the opening of the angle. For an angle whose vertex is outside the circle, two arcs are involved.
The larger, farther arc must be considered before the smaller, nearer arc. Drawing a light mark on both arcs can prevent reversed subtraction.
Segment product rules are not isolated facts to memorise. They are consequences of similar triangles. When two lines cut through a circle, the angles made by the lines and the angles that face matching arcs create triangles with the same shape.
Similar triangles have proportional side lengths. Rearranging those proportions produces the equal products used in chord and secant problems.
This explains why a segment from outside the circle is paired with the entire line segment through the circle, not just the part inside. The whole distance begins at the external point and ends at the far intersection.
Power of a point gives a fixed value for every line drawn from one chosen point through a circle. A point inside the circle produces products from the two pieces of a chord. A point outside produces a product involving an outside part and a whole secant.
A tangent gives another way to find that same value. This idea is useful in coordinate geometry, where a circle and a line may be represented by equations. It is relevant in design work too, since curved paths, wheels, circular windows, and round structures often need accurate distances from lines to a circular boundary.
A reliable solution method starts by sorting the information into angle, arc, or length facts. Mark the centre, all points where a line enters or leaves the circle, and any point where a line only touches the circle. Then decide where the vertex of the target angle lies, or where the two relevant lines meet.
Keep arc measures separate from angle measures in your working. For length problems, write every segment from the same outside point in order, using outside length first and whole length second.
Check whether an answer is sensible. An arc cannot exceed three hundred sixty degrees, a minor arc is less than one hundred eighty degrees, and a distance cannot be negative.