MYP Standard Mathematics 3.3 Circle Theorems Notes

Circle Language Lets You Describe Geometry Precisely

1. Circles appear simple, but a lot of powerful geometry comes from naming their parts accurately and using a small set of reliable facts (the circle theorems).
2. In this section you will review the key parts of a circle and then build the essential angle and tangent properties used in problem solving.

Tangents Touch At One Point And Create A Right Angle With The Radius

A tangent to a circle touches the circle at exactly one point (the point of contact).

A key property links tangents to the circle's center:

Tangent–radius theorem: If a line is tangent to a circle at point $T$ and $OT$ is the radius to the point of contact, then $$OT \perp \text{tangent at }T$$

In other words, the tangent is perpendicular to the radius drawn to the contact point.

Angles At The Center Are Twice Angles At The Circumference

Many circle problems involve angles subtended by the same chord or arc.

Central angle theorem: For points $A,B,C$ on a circle with center $O$ (and with $B$ on the same side of chord $AC$ as the arc being used), $$\angle AOC = 2\angle ABC$$

So, the angle at the center is double the angle at the circumference standing on the same arc $AC$.

The Angle In A Semicircle Is A Right Angle

If $AC$ is a diameter, then any point $B$ on the circle gives a right angle at $B$.

Angle in a semicircle theorem: If $AC$ is a diameter, then $$\angle ABC = 90^\circ$$

This follows from the central angle theorem: the angle subtended by diameter $AC$ at the center is $\angle AOC=180^\circ$, so the angle at the circumference is half of that.

Angles In The Same Segment Are Equal

If two angles stand on the same chord (or the same arc), they are equal.

Same segment theorem: If $A,B,C,D$ lie on the circumference and chord $AC$ is fixed, then $$\angle ABC = \angle ADC$$ provided both angles subtend chord $AC$ from the same side.

This is an immediate consequence of the central angle theorem: both angles are half of the same central angle $\angle AOC$.

Opposite Angles In A Cyclic Quadrilateral Sum To 180°

A cyclic quadrilateral is a four-sided shape whose vertices all lie on a circle.

Cyclic quadrilateral theorem: If $A,B,C,D$ lie on a circle, then $$\angle ABC + \angle ADC = 180^\circ,$$ and similarly the other opposite pair also sum to $180^\circ$.

The idea is that these two angles stand on complementary arcs whose central angles add to $360^\circ$, so the corresponding circumference angles add to $180^\circ$.

Tangent–Chord Angles Link Circle Geometry To Straight Lines

A powerful tangent fact connects an angle made with a tangent to an angle in the circle.

Alternate segment theorem: If a tangent touches the circle at $A$ and chord $AB$ is drawn, then the angle between the tangent and chord $AB$ equals the angle in the opposite segment, for example $$\angle(\text{tangent at }A,\ AB)=\angle ACB,$$ where $C$ is a point on the circle on the opposite side of chord $AB$.

This theorem is frequently combined with parallel lines to prove triangles are isosceles or to show two triangles are similar.

Problem-Solving Patterns From Typical Proof Questions

Many circle-theorem proofs (including those like the practice problems in your course) rely on the same small set of moves.

Pattern 1: Convert An Angle Between Tangents Into A Central Angle

1. If tangents at $X$ and $Y$ meet at $Q$, then radii $OX$ and $OY$ are perpendicular to the tangents, so quadrilateral $OXQY$ has two right angles.
2. This gives a relationship between $\angle XQY$ and the central angle $\angle XOY$, and then you can use "center is twice circumference" to relate to an angle like $\angle XPY$.

Pattern 2: Use Cyclic Quadrilaterals To Prove Similarity

1. If two lines are extended to meet outside a circle (for example, chords extended to meet at $X$), angle equality often comes from:
   1. angles in the same segment, and
   2. vertically opposite angles at the external intersection.
2. Once you have two equal angles, triangle similarity usually follows.

Pattern 3: Prove Isosceles By Showing Base Angles Are Equal

Triangles inside circle problems become isosceles when you can prove two angles are equal. Common reasons:

1. radii create isosceles triangles (two radii are equal),
2. same-segment angles are equal, or
3. alternate segment theorem converts a tangent angle into a circumference angle.

Converse Statements Matter In Geometry Proofs

1. Many theorems are written in the form "If $p$, then $q$."
2. The converse swaps them: "If $q$, then $p$." For example:
   1. Theorem: If $AC$ is a diameter, then $\angle ABC=90^\circ$.
   2. Converse: If $\angle ABC=90^\circ$, then $AC$ is a diameter.
3. In circle geometry, converses are often true and are extremely useful for proving points lie on a circle (or that a line is tangent).