Circles, Lines, and Segments

A circle is a set of points equidistant from a center point. When you see "circle P," that means P is the center.

There are four key segments and lines that interact with circles:

• A radius connects the center to any point on the circle
• A chord is a segment with both endpoints on the circle (a diameter is a special chord that passes through the center)
• A secant is a line that crosses the circle at two points
• A tangent is a line that touches the circle at exactly one point (called the point of tangency)

Quick Tip: Think of tangent lines as "just barely touching" the circle, while secants "cut through" the circle completely.

When two circles exist in the same plane, they can be:

• Intersecting at two points
• Tangent to each other (touching at one point)
• Completely separate
• Concentric (sharing the same center)

Circles can also share common tangent lines—either external tangents (not passing between the circles) or internal tangents (passing between them).

Tangent Line Properties

The relationship between tangent lines and radii is special and extremely useful. The Tangent Line to Circle Theorem states that a line is tangent to a circle if and only if it's perpendicular to the radius at the point of tangency.

This makes perfect sense when you think about it: the radius points "straight out" from the circle, and the tangent line runs "straight across" at that point.

Another important property is the External Tangent Congruence Theorem. When you draw two tangent segments from a point outside a circle, those segments are always equal in length.

Remember: Whenever you see a line tangent to a circle, there must be a radius perpendicular to it at the point of contact. This relationship is guaranteed!

These properties help us solve many geometry problems involving circles and tangent lines, especially when we need to find missing angles or lengths.

Properties of Chords

Chords have several important properties that help us solve circle problems:

The Congruent Corresponding Chord Theorem tells us that in the same circle (or in congruent circles), two minor arcs are congruent if and only if their corresponding chords are congruent. Simply put: equal chords create equal arcs, and vice versa.

When a diameter is perpendicular to a chord, something special happens. The Perpendicular Chord Bisector Theorem states that the diameter will bisect (cut in half) both the chord and the arc it creates. The converse is also true: if one chord perpendicularly bisects another, then the first chord must be a diameter.

Make this practical: When you need to find the exact center of a circle, draw two chords and find their perpendicular bisectors. The point where they intersect is the center!

The Equidistant Chord Theorem gives us another useful relationship: two chords are congruent if and only if they're equidistant from the center. This means equal chords are always the same distance from the center of the circle.

Inscribed Angles and Polygons

When angles and polygons are inscribed in a circle, they follow predictable patterns.

The Inscribed Right Triangle Theorem gives us a way to identify right angles: if a right triangle is inscribed in a circle, its hypotenuse must be a diameter. The reverse is also true—if one side of an inscribed triangle is a diameter, the angle opposite it must be 90°.

Quadrilaterals can be inscribed in circles too. The Inscribed Quadrilateral Theorem tells us that a quadrilateral can fit perfectly on a circle if and only if its opposite angles are supplementary (add up to 180°).

Test Tip: On exams, look for problems with inscribed shapes. If you spot a quadrilateral inside a circle, immediately check if opposite angles add to 180°!

You can use these properties to solve for missing angle measures. For example, if you know a triangle is inscribed in a circle with one side as a diameter, you can immediately identify the opposite angle as 90°, which makes solving for other angles much easier.

Circles in the Coordinate Plane

Circles can be defined using coordinate geometry with some straightforward equations.

The simplest case is a circle centered at the origin (0,0). For any point (x,y) on this circle, the equation is x² + y² = r², where r is the radius. This comes directly from the Pythagorean theorem, since the distance from any point to the origin must equal the radius.

For circles centered at any point (h,k), the equation becomes $x-h$² + $y-k$² = r². This is called the standard equation of a circle. The $x-h$ and $y-k$ terms represent the horizontal and vertical distances from any point (x,y) on the circle to the center.

Pro Strategy: When solving circle problems in the coordinate plane, first identify the center and radius, then plug them into the standard equation.

Finding the equation is straightforward if you know the center and radius. If you know the center and a point on the circle, you can find the radius by calculating the distance between these points, then form the equation.