Notes Circle Basics M RECALL EXAMPLES Give an example of each of the following: 1. Name the circle 2. Radius 3. Chord 4. Diameter 5. Secant 6. Tangent (line) 7. Point of tangency 8. Tangent (segment) DEFINTION A of a circle is an angle whose vertex is the center of the circle. Minor Arcs Major Arcs < 180 > 180 Named by endpoints of the arc Named by endpoints and one other point A is an arc with endpoints that are the endpoints of a diameter. EXAMPLES Determine if each is a minor arc, major arc, or semicircle. Find the measure of each. 9. RS 10. RTS 11. RST
12. Name a major arc. 13. Name a minor arc. POSTULATE Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. mabc = mab + mbc EXAMPLES Identify the given arc as a major arc, minor arc, or semicircle, then find the measure of each arc. 14. TQ 15. QRT 16. TQR 17. QS 18. TS 19. RST CONGRUENT CIRCLES Two circles are congruent if they have the same radius CONGRUENT ARCS Two arcs are congruent if they have the same measure and they are arcs of the same circle or of congruent circles. EXAMPLES Are the arcs congruent? Explain. 20. 21.
WKS Circle Basics M SHOW ALL WORK!
Notes Inscribed Angles & Polygons DEFINITION An angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the arc of the angle. Th THEOREM The measure of an inscribed angle is one half the measure of its intercepted arc. m ADB = 1 2 mab OR mab = 2 m ADB EXAMPLES Find the indicated measure. 1. m T 2. mrs 3. mqr 4. m STR THEOREM If two inscribed angles of a circle intercept the same arc, then the angles are congruent. EXAMPLES Find the indicated measure. 5. m HGF 6. mtv 7. m ZXW
DEFINITION A polygon is an polygon if all The circle is said to the of its vertices lie on a circle. polygon. THEOREM A right triangle is inscribed in a circle if and only if the hypotenuse is a diameter of the circle. THEOREM A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. EXAMPLES Find the value of each variable 1. 2. 3. 4.
WKS Inscribed Angles & Polygons Find the measure of each variable, given arc, or given angle. SHOW ALL WORK! 1. 2. 3. Th 4. 5. 6. 7. 8. 9. 10. 11.
Notes Properties of Chords M What is a chord? NOTE Any chord divides a circle into two arcs, the major arc and the minor arc. A diameter divides a circle into two semicircles. THEOREM In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. AB CD if and only if AB CD EXAMPLES 1. Find mfg. 2. If mab = 110, find mbc. 3. If mac = 150, find mab. Find the measure of each arc of a circle circumscribed around the following regular polygons. 4. Square 5. Hexagon 6. Nongon
THEOREMS Diameters, Chords, and Perpendicular Bisectors If one chord is a perpendicular bisector of another, then the first is a diameter. If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. EXAMPLES CF = 7. Find the measure of each arc or segment. 7. EF 8. EC 9. CD F 10. DE 11. CE THEOREM In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. (perpendicular to radius) AB CD if and only if EF = EG EXAMPLES Find the indicated values 12. CU 13. QU 14. The radius of C
WKS Chords M Find the indicated measure. 1. mab 2. mab 3. EG Solve for x. 4. 5. 6. 7. 8. *Note* 3x + 2 = ½ the chord AD 9.
What can you conclude about the diagram shown? 10. 11. 12. 13. In the diagram of R, which congruence relationship is not necessarily true? Find the measure of each arc of a circle circumscribed about the regular polygon. 14. Triangle 15. Pentagon 16. Octagon
Notes Tangents W RECALL A tangent is any line or segment that touches the edge of a circle in exactly one spot. THEOREM In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle. Line m is tangent to Q if and only if m QP Examples 1. Given that PT is a radius, determine if ST is tangent to the circle. 2. Find the value of r so that AB is tangent to the circle and CB is a radius.
THEOREM Tangent segments from a common external point are congruent. (Ice Cream Cone Theorem) If ST and SR are tangents, then ST SR. EXAMPLES Find the value of x. 3. 4.
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Notes More Angles THEOREM If a tangent and a chord intersect at a point on a circle, then the measure of each angle is one half the measure of intercepted arc. Th Examples 1. m 1 2. mrst 3. mxy THEOREM If an angle is on the outside of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. m 1 = 1 (mbc mac ) 2 m 2 = 1 (mpqr mpr ) 2 m 3 = 1 (mxy mwz ) 2 Examples 4. 5. 6.
7. 8.
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Notes Equations of Circles F EQUATION The equation of a circle with center (h, k) and a radius of r units is: (x h) 2 + (y k) 2 = r 2 Examples Write an equation for the circle with the given center and radius. Then graph the circle. 1. Center at (0, 0) and r = 1 2. Center at (2, -4) and r = 1 Find the circle s center and radius, then graph the equation. 3. (x + 1) 2 + (y 2) 2 = 9 4. (x 2) 2 + (y 1) 2 = 4 5. x 2 + y 2 = 16
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