Lines that intersect Circles

Lines that intersect Circles Geometry H2 (Holt 12-1) K. Santos

Circle definition Circle: set of all points in a plane that are a given distance (radius) from a given point (center). Circle P P Radius: is a segment that connects the center of the circle to a point on the circle

Interior & Exterior of a circle Interior of a circle: set of all the points inside the circle Exterior of a circle: set of all points outside the circle

Lines & Segments that intersect a circle A G O B F E C D Chord: is a segment whose endpoints lie on a circle. Diameter: -a chord that contains the center -connects two points on the circle and passes through the center Secant: line that intersects a circle at two points

Tangent • A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point • Tangent may be a line, ray, or segment • The point where a circle and a tangent intersect is the point of tangencyA B

Pairs of circles Congruent Circles: two circles that have congruent radii Concentric Circles: coplanar circles with the same center

Tangent Circles Tangent Circles: coplanar circles that intersect at exactly one point Internally tangent externally tangent circles circles

Common Tangent Common tangent: a line that is tangent to two circles Common external common internal tangents tangents

Theorem 12-1-1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. O A P B Given: is tangent to circle O Then:

Example is tangent to circle O. Radius is 5” and ED = 12” Find the length of . O E D

Example Find x. 130 x

Theorem 12-1-2 If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. O A P B Given: Then: is tangent to circle O

Example—Is there a tangent line? Determine if there is a tangent line? 12 6 8

Theorem 12-1-3 If two segments are tangent to a circle from the same point, then the segmentsare congruent. A B C Given: and are tangents to the circle Then:

Example: R and are tangent to circle Q. 2n – 1 n + 3 Find RS. T S

Lines that intersect Circles