14-2b Tangent Lines to Circles

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Presentation transcript:

14-2b Tangent Lines to Circles Proof Geometry

Center of Circle To find the center of the circle through a construction 1) Inscribe a triangle in the circle 2) Construct the perpendicular bisectors of each of the sides of the triangle

Circle Center and Chord Theorems: The Perpendicular from the center of the circle to a chord bisects the chord. The segment from the center of the circle to the midpoint of a chord is perpendicular to the chord. The perpendicular bisector of a chord passes through the center of the circle.

Circle Center and Chord Theorems: Sketch out the proof for: The Perpendicular from the center of the circle to a chord bisects the chord. 1)Create a circle with center P and a chord QR. Let PF be perpendicular to chord QR. 2) Introduce radii PQ and PR. 3) Show the two smaller triangles congruent through HL.

Circle Center and Chord Theorems: Sketch out the proof for: The segment from the center of the circle to the midpoint of a chord is perpendicular to the chord. 1)Create a circle with center P and a chord QR. Let PF be the bisector of chord QR. 2) Introduce radii PQ and PR. 3) Show the two smaller triangles congruent through SSS.

Circle Center and Chord Theorems: 1)Create a circle with center P and a chord QR. Let m be the perpendicular bisector of QR through F. 2) By the perpendicular bisector theorem, any point on m is equidistant from the endpoints. 3) Uniqueness shown through any point equidistant from endpoints is on the perpendicular bisector (Thm 6-4). Circle Center and Chord Theorems: Sketch out the proof for: The perpendicular bisector of a chord passes through the center of the circle.

Equidistant Chords Theorem: In the same circle or in congruent circles, chords are equidistant from the center if and only if they are congruent.

Equidistant Chords Theorem: Show: IF equidistant from the center THEN congruent.

Equidistant Chords Theorem: Show: IF congruent THEN equidistant from the center.

Homework Pg. 457 #16-18 Pg. 461 #1, 5, 8