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Lines that intersect Circles

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

2 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

3 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

4 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

5 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 tangency A B

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

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

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

9 Theorem 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: 𝐴𝐵 is tangent to circle O Then: 𝐴𝐵 ⟘ 𝑂𝑃

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

11 Example Find x. 130° x

12 Theorem 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: 𝐴𝐵 ⟘ 𝑂𝑃 Then: 𝐴𝐵 is tangent to circle O

13 Example—Is there a tangent line?
Determine if there is a tangent line?

14 Theorem If two segments are tangent to a circle from the same point, then the segments are congruent. A B C Given: 𝐴𝐵 and 𝐶𝐵 are tangents to the circle Then: 𝐴𝐵 ≅ 𝐶𝐵

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

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