Chapter 14: CIRCLES!!!Proof Geometry

DefinitionsA circle is the set of all points of a plane that are equidistant from a single point.Picture:

A sphere is the set of all points in space that are equidistant from a single point.Picture:

DefinitionsCircles (or spheres) that share the same center are called concentric.Picture:

Parts of a circleA chord: A segment whose endpoints lie on the circle.

A secant: A line which intersects the circle in two points.

Every chord determines a secant and every secant contains a chord

A diameter is a chord containing the center of the circle.

A radius is a segment from the center to a point of the circle. The point is called the outer end of the radius.

Recall:What’s a tangent again?

A tangent to a circle is a line which intersects the circle in one and only one point.

This point is called the point of tangency or the point of contact.

We say that the line and the circle are tangent at this point.

Tangent Perpendicular to Radius Theorem

A line is perpendicular to a radius at its outer end if and only if it is tangent to the circle.

Definitions: Interior and Exterior

A point is on the interior of a circle if the distance from the center is less than the radius.

A point is on the exterior of a circle if the distance from the center is greater than the radius.

Can 2 circles be tangent to one another?

• Two circles are tangent, if they are tangent to the same line at the same point.• Internally tangent• Centers on same side of tangent line

• Externally tangentCenters on opposite sides of tangent line

Example (#12 on p. 457)

Prove that if two circles having congruent radii are externally tangent, then any point equidistant from their centers is on their common tangent.

Homework

pg. 452: # 2, 5Pg. 456 #4, 5, 9, 11, 13, 14