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Chapter 10

Chapter 10. Circles. 10.1 Circles and Circumference. Circle – points all equidistant from a center point Center – the foci of a circle, used to name the circle but not part of the circle Chord – any segment whose endpoint are on the circle

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Chapter 10

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  1. Chapter 10 Circles

  2. 10.1 Circles and Circumference • Circle – points all equidistant from a center point • Center – the foci of a circle, used to name the circle but not part of the circle • Chord – any segment whose endpoint are on the circle • Radius – the distance from the center to any point on the circle • Diameter – A chord that passes through the center of a circle, also twice the radius • Circumference – the distance around the circle • Pi (π) – the ratio of C/d. 3.14159…

  3. Things to Know • C = 2πr • D = 2r

  4. 10.2 Angles and Arcs • Central Angle – has center of circle as vertex and two sides as radii • Arc = part of the circle separated by a angle. • Minor Arc – An arc which is less than 180 degrees • Major Arc – An arc which is more than 180 degrees • Semicircle - half a circle

  5. Key Concepts • The sum of all central angles must equal 360 degrees

  6. Theorems • In the same or congruent circles, two arcs are congruent if and only if their corresponding central angles are congruent. • The measure of arc formed by two adjacent arcs is the sum of the measures of two arcs

  7. 10.3 Arcs and Chords • Inscribed – set inside a circle with all vertexes on the circle • Circumscribed – Set around a polygon with all vertexes on the circle

  8. Theorems • In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent • In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the cord and its arc. • In a circle or in congruent circles, two chords are congruent if and only if they are equidistant form the center

  9. 10.4 Inscribed Angles • Intercepted Angle – an angle which has its vertex on the circle and two chords forming its side • Intercepted Arc – the resulting arc from and intercepted angle

  10. Theorems • If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc (or the measure of the intercepted are is twice the measure of the inscribed angle) • If two inscribed angles of a circle (or two congruent circles) intercept congruent arcs or the same arc, then the angles are congruent.

  11. More Theorems • If an inscribed angle intercepts a semicircle, the angle is a right angle. • If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

  12. 10.5 Tangents • Tangent: line Intersecting at one point • Point of Tangency: The point of intersection between the tangent and circle

  13. Theorems to know • If a line is tangent to a circle, then it is perpendicular to the radius drawn to point of tangency. • If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. • If two segments from the same exterior point are tangent to a circle, then they are congruent.

  14. 10.6 Scants, Tangents, and Angle Measures • Scant – a line that intercepts a circle in exactly two points.

  15. Theorems • If two secants intersect in the interior of a circle, then he measure of an angle formed is one-half the sum of the measure of the arcs intercepted by the angle and its vertical angle • If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed in one-half the measure of its intercepted arc.

  16. More Theorems

  17. 10.7 Special Segments in a Circle • If two chords intersects in a circle, then the products of the measures of the segments of the chords are equal • If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.

  18. More Theorems • If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.

  19. 10.8 Equations of Circles • An equation for a circle with center at (h,k) and radius of r units is • (x – h)2 + (y – k)2 = r2

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