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Day 70: Circles

Day 70: Circles. The Return. Recap. Radius Diameter Circumference Pi Secant Tangent Chord Area Central Angle Arc Arc Measure Arc Length Sector Area of a Sector. Inscribed Angles. An inscribed angle has its vertex on the circle, and its rays intersect the circle

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Day 70: Circles

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  1. Day 70:Circles The Return

  2. Recap • Radius • Diameter • Circumference • Pi • Secant • Tangent • Chord • Area • Central Angle • Arc • Arc Measure • Arc Length • Sector • Area of a Sector

  3. Inscribed Angles • An inscribed angle has its vertex on the circle, and its rays intersect the circle • ACB is an inscribedangle. • The arc formed by aninscribed angle is calledan intercepted arc. • AB is an intercepted arc

  4. Inscribed Angle Theorem • The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. • mB = ½(arc AC) • arc AC = 2(mB) • mC = ½(arc AB) • arc AB = 2(mC) • mA = ½(arc BC) • arc BC = 2(mA)

  5. Inscribed Angle Theorem • Example: • If mADB = 24, whatis the measure of AB? • What is mACB? • If AB = 36, whatis mADB? • What is mACB?

  6. Inscribed Angle Theorem Proof • Given: B inscribed inסּO • Prove: mB = ½ AC • First we construct radius OC. • This makes isosceles triangleBOC. • mAOC = mB + mC (exteriorangle) • Since mB = mC, then mAOC = 2mB. • B = ½ mAOC = ½ AC B O C A

  7. Inscribed Angle Theorem Corollary • If two angles inscribe the same arc, or congruent arcs, then the angles are congruent. • Example: B and D bothinscribe AC. • Therefore, B  D B D O C A

  8. Inscribed Angle Theorem Corollary • We previously demonstrated that the arc measure of a semicircle, such as PRQ, is 180. • What is the measureof an angle inscribed bya semicircle (e.g. PRQ)? • An inscribed angle of atriangle intercepts adiameter or semicircle if andonly if it is a right triangle.

  9. Inscribed Quadrilateral Theorem • A quadrilateral can be inscribed inside of a circle if and only if its opposite angles are supplementary. • Example: mA + mC = 180and mB + mD = 180 • Proof: B and D inscribeadjacent arcs that create thewhole circle. Since the two arcsadd to make 360, and the angles are half of the arcs, then the two angles add to 180 C B O D A

  10. Tangent/Radius Theorem • Any tangent of a circle is perpendicular to a radius of that circle at their point of intersection. • Indirect Proof • If a diameter intersectsa tangent at each end,those tangents areparallel.

  11. Intersecting Tangents • If two segments from the same exterior point are tangent to a circle, then those segments are congruent. • AP and AQ are tangent tothe circle. • AP  AQ. P Q A

  12. Homework 43 • Workbook, pp. 129, 131

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