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CONFIDENTIAL 1 Geometry Lines That Intersect Circles

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CONFIDENTIAL 2 Warm Up A Point is chosen randomly on. Use the diagram to find the probability of each event. 1) The point is not on 2) The point is on 3) The point is or 4) The point is on 1) ¾2) ½3) 13/204) 1/40

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CONFIDENTIAL 3 Lines That Intersect Circles A Circle is the set of all points in a plane that are equidistant from a given point, called the center of the circle. A circle with center C is called circle C, or C. The Interior of a circle is the set of all points inside the circle. The exterior of a circle is the set of all points outside the circle.

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CONFIDENTIAL 4 Lines and Segments That Intersects circles TERMDIAGRAM A Chord is a segment whose end points lie on a circle. A Secant is a line that intersects a circle at two points. A Tangent is a line in the same plane as a circle that intersects it at exactly one point. The point where the tangent and a circle intersect is called the point of tangency.

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CONFIDENTIAL 5 Identifying Lines and Segments That Intersect Circles chords: and tangent: L radii: and secant: diameter: Identify each line or segment that intersect A.

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CONFIDENTIAL 6 Now you try Identify each line or segment that intersects P. chord: QR tangent: UV radii: SP and PT secant: ST diameter: ST

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CONFIDENTIAL 7 Remember that the terms radius and diameter may refer to line segments, or to the lengths of segments

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CONFIDENTIAL 8 TERMCONDITION Two circles are congruent circles if and only if they have congruent radii. Concentric circles are coplanar circles with the same center.

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CONFIDENTIAL 9 Two coplanar circles that intersect at exactly one point are called tangent circles

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CONFIDENTIAL 10 Identifying Tangents of a circles Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. A B X Y -4 4 0 -2 2

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CONFIDENTIAL 11 A B X Y -4 4 0 -2 2

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CONFIDENTIAL 12 Now you try Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point 0-4 4 4 Point of tangency: (2, -1) equation of the tangent: y = -1

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CONFIDENTIAL 13 A common tangent is a line that is tangent to two circles

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CONFIDENTIAL 14 B A p q

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CONFIDENTIAL 15 Construction Tangent to a circle at a point 1) 2)

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CONFIDENTIAL 16 3) l

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CONFIDENTIAL 17 The tangent line is perpendicular to the radius at the point of tangency. This fact is the basis for the following theorems.

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CONFIDENTIAL 18 THEOREMHYPOTHESISCONCLUSION 1) 2) A B l

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CONFIDENTIAL 19 THEOREMHYPOTHESISCONCLUSION 3)

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CONFIDENTIAL 20 Using Properties of Tangents

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CONFIDENTIAL 21 Now you try 1) R S T X- 6.3 RS = 2.1

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CONFIDENTIAL 22 2) S T R n+3 2n-1 Now you try RS = 7

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CONFIDENTIAL 23 Now some problems for you to practice.

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CONFIDENTIAL 24 ASSESSMENT Identify each line or segment that intersects each circle. 1) F D E m l 2) T Q S

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CONFIDENTIAL 25 Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. X Y 0 A B -4 4 3) Point of tangency: (-1, 4) equation of the tangent: y = 4

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CONFIDENTIAL 26 Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. X Y 0 R S -4 4 4) length of radius OR = 2 Point of tangency: (-1, 0) equation of the tangent: y = 0 length of radius OS = 2 Point of tangency: (2, 0) equation of the tangent: y = 0

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CONFIDENTIAL 27 K C L J 4x-1 2x+9 5) The segments in each figure are tangent to the circle. Find length JK. JK = 7

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CONFIDENTIAL 28 T P U S Y- 4 3434 y The segments in each figure are tangent to the circle. Find length ST. 6) ST = 12

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CONFIDENTIAL 29 Lines and Segments That Intersects circles TERMDIAGRAM A Chord is a segment whose end points lie on a circle. A Secant is a line that intersects a circle at two points. A Tangent is a line in the same plane as a circle that intersects it at exactly one point. The point where the tangent and a circle intersect is called the point of tangency. REVIEW

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CONFIDENTIAL 30 Identifying Lines and Segments That Intersect Circles Identify each line or segment that intersect A. chords: and tangent: L radii: and secant: diameter:

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CONFIDENTIAL 31 TERMCONDITION Two circles are congruent circles if and only if they have congruent radii. Concentric circles are coplanar circles with the same center.

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CONFIDENTIAL 32 Two coplanar circles that intersect at exactly one point are called tangent circles

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CONFIDENTIAL 33 Identifying Tangents of a circles Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. A B X Y -4 4 0 -2 2

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CONFIDENTIAL 34 A B X Y -4 4 0 -2 2

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CONFIDENTIAL 35 Now you try Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point 0-4 4 4

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CONFIDENTIAL 36 A common tangent is a line that is tangent to two circles

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CONFIDENTIAL 37 B A p q

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CONFIDENTIAL 38 THEOREMHYPOTHESISCONCLUSION 1) 2) A B l

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CONFIDENTIAL 39 THEOREMHYPOTHESISCONCLUSION 3)

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CONFIDENTIAL 40 Using Properties of Tangents

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CONFIDENTIAL 41 You did a great job today!

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