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Lines that intersect Circles

Lines that intersect Circles. Geometry CP2 (Holt 12-1) K. Santos. Circle definition. Circle: points in a plane that are a given distance (radius) from a given point (center). Circle P P Radius: center to a point on the circle. Interior & Exterior of a circle.

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Lines that intersect Circles

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  1. Lines that intersect Circles Geometry CP2 (Holt 12-1) K. Santos

  2. Circle definition Circle: points in a plane that are a given distance (radius) from a given point (center). Circle P P Radius: center to a point on the circle

  3. Interior & Exterior of a circle Interior of a circle: points inside the circle Exterior of a circle: points outside the circle exteriorinterior

  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 • The point where a circle and a tangent intersect is the point of tangencyA Point B 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. Theorems 12-1-1 & 12-1-2 Radius perpendicular to tangent line (at point of tangency) O A P B

  10. Example is tangent to circle O. Radius is 5” and ED = 12” Find the length of . O E D Remember Pythagorean theorem (let = x) = + = 25 + 144 = 169 x= x = 13

  11. Example—Is there a tangent line? Determine if there is a tangent line? 12 6 8 If there is a tangent then there must have been a right angle (in a right triangle). Test for a right angle. + 144 36 + 64 144 100 so there is no right angle, no tangent line

  12. Example Find x. 130 x Radius tangents (right angles) Sum of the angles in a quadrilateral are 360 90 + 90 + 130 = 310 x = 360 - 310 x = 50

  13. Theorem 12-1-3 Same external point tangents congruent A B C Then:

  14. Example: R and are tangent to circle Q. 2n – 1 n + 3 Find RS. T S RT = RS 2n – 1 = n + 3 n – 1 = 3 n = 4 RS = n + 3 RS = 4 + 3 RS = 7

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