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Arc Measures

This lesson focuses on the properties of circles, including central and inscribed angles, and the relationship between arcs and the area of sectors. Students will apply these concepts to solve problems related to arc lengths and sector areas, using examples from surveys and circle graphs. Key concepts include the measurement of minor and major arcs, along with the identification of congruent arcs. By the end of the lesson, students will be equipped to analyze and solve related problems effectively.

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Arc Measures

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  1. Arc Measures MM2G3. Students will understand the properties of circles. b. Understand and use properties of central, inscribed, and related angles. c. Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Standards:

  2. Find the measure of each arc of P, where RTis a diameter. a. b. RTS RS c. RST a. RSis a minor arc, so mRS=mRPS=110o. b. – RTSis a major arc, so mRTS = 360o 110o = 250o. c. RT is a diameter, so RSTis a semicircle, and mRST=180o. EXAMPLE 1 SOLUTION

  3. Survey A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. mAC a. mAB mBC mAC a. = + EXAMPLE 2 SOLUTION = 29o + 108o = 137o

  4. Survey A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. mACD b. b. mACD = mAC + mCD EXAMPLE 2 SOLUTION = 137o + 83o = 220o

  5. Survey A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. c. mADC c. – mAC mADC = 360o EXAMPLE 2 SOLUTION = 360o – 137o = 223o

  6. 3 . TQR TQRis a semicircle, so m TQR =180o. 4 . QS QSis a minor arc, so m QS =160o. GUIDED PRACTICE SOLUTION SOLUTION

  7. 5 . TS TSis a minor arc, so m TS =80o. 6 . RST RSTis a semicircle, so m RST =180o. GUIDED PRACTICE SOLUTION SOLUTION

  8. In the diagram, PQ, FGJK, and mJK= 80o. Find mFG Because FGand JKare congruent chords in congruent circles, the corresponding minor arcs FGand JKare congruent. So, mFG = mJK = 80o. EXAMPLE 3 Use congruent chords to find an arc measure SOLUTION

  9. Use the diagram of D. 1. If mAB= 110°, find mBC ANSWER mBC = 110° GUIDED PRACTICE

  10. Use the diagram of D. 2. If mAC = 150°, find mAB ANSWER mAB = 105° GUIDED PRACTICE

  11. a. b. a. CDEF because they are in the same circle and mCD = mEF b. RSand TUhave the same measure, but are not congruent because they are arcs of circles that are not congruent. EXAMPLE 4 Identify congruent arcs Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

  12. c. VXYZ because they are in congruent circles and mVX=mYZ . c. EXAMPLE 5 Identify congruent arcs Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

  13. ABCD because they are in congruent circles and mAB=mCD . GUIDED PRACTICE Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

  14. 8. MNand PQhave the same measure, but are not congruent because they are arcs of circles that are not congruent. GUIDED PRACTICE Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

  15. Vocab:

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