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Day 71: Circles

Day 71: Circles. Circling the drain. Intersecting Chords. If two chords (or secants) intersect inside a circle, then the measure of the angle formed by the chords is half of the sum of the measures of the two arcs intersected by the chords. QBS = ½(QS + TR). Intersecting Chords Proof.

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Day 71: Circles

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  1. Day 71:Circles Circling the drain

  2. Intersecting Chords • If two chords (or secants) intersect inside a circle, then the measure of the angle formed by the chords is half of the sum of the measures of the two arcs intersected by the chords. • QBS = ½(QS + TR)

  3. Intersecting Chords Proof • Construct RS to make BSR • m1 = m2 + m3 (Ext. ) • m2 = ½mQSm3 = ½mTR(An inscribed  is ½ the arc) • By substitution: m1 = ½mQS + ½mTR • And by factoring: m1 = ½(mQS + mTR) 3 1 2

  4. Intersecting Chords • If two chords (or secants) intersect inside a circle, then the products of the lengths of the chord segments are equal. • QB∙BR = SB∙BT

  5. Intersecting Chords Proof • Again, we can create triangles. • QBT  SBR (vertical s) • Q  S (inscribed sintersecting the same arc) • QBT  SBR (AA) • (CSSTP) • QB∙BR = SB∙BT (cross-x)

  6. Intersecting Secants • If two secants of a circle intersect outside of that circle, then the measure of the angle formed by the secants is half of the difference of the measures of the arcs intercepted by the secants. • E = ½(DF – MN)

  7. Intersecting Secants Proof • Create EDN • m2 = m1 – m3 (Ext. ) • m1 = ½mDFm3 = ½mMN(An inscribed  is ½ the arc) • By substitutionm2 = ½mDF – ½mMN • By factoring m2 = ½(mDF – mMN) 2 3 1

  8. Intersecting Secants • An external secant segment is a secant segment that lies in the exterior of a circle. (eg. ME and NE) • If two secants intersect outside of a circle, then the product of one segment length and its external secant segment length is equal to the product of the other secant length and its external secant segment length. • EN∙EF = EM∙ED

  9. Intersecting Secants Proof • Construct ND and MF • D  F (inscribed s intersecting the same arc) • EDN  EFM (they share E; AA) • (CSSTP) • EN∙EF = EM∙ED (cross-multiply)

  10. Examples • If RS = 60 and NO = 20, find the measures of NPO, NPR, and M. • If MN = 5, NR = 3,and OS = 6, find MO. • If SP = 7, NP = 3,and RP = 9,find OP.

  11. Chord/Tangent • If a tangent intersects a chord, the measure of the angle formed is half of the measure of the arc formed. • mAPQ = ½PQ R P Q A

  12. Secant/Tangent • If a secant and a tangent intersect outside the circle, it is very similar to two secants. • mA = ½(PR – PQ) R P Q A

  13. Tangent/Tangent • Two tangents intersecting is similar to two secants intersecting. • A = ½(PRQ – PQ) R P Q A

  14. Summary of Angles • If the angle is on the circle, it is half of the intercepted arc. • If the angle is inside the circle, it is half the sum of the two arcs. • If the angle is outside the circle, it is half the difference between the two arcs.

  15. Secant/Tangent • If a secant and a tangent intersect outside the circle, it is very similar to two secants. • The tangent is both the‘exterior segment’ and the ‘whole segment’. • AP∙AP = AQ∙AR or AP2 = AQ∙AR R P Q A

  16. Homework 44 • Workbook, pp. 134, 136

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