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Warm up

Warm up. Find the value of each variable. a. b. 32 °. c. Applications: Motion in Circles. We say an object rotates about its axis when the axis is part of the moving object. A child revolves on a merry-go-round because he is external to the merry-go-round's axis.

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Warm up

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  1. Warm up • Find the value of each variable. a b 32° c

  2. Applications: Motion in Circles • We say an object rotates about its axis when the axis is part of the moving object. • A child revolves on a merry-go-round because he is external to the merry-go-round's axis.

  3. Applications: Centrifugal Force • Although the centripetal force pushes you toward the center of the circular path... • ...it seems as if there also is a force pushing you to the outside. This apparent outward force is called centrifugal force. • We call an object’s tendency to resist a change in its motion its inertia. • An object moving in a circle is constantly changing its direction of motion.

  4. Applications: Centrifugal Force • This is easy to observe by twirling a small object at the end of a string. • When the string is released, the object flies off in a straight line tangent to the circle. • Centrifugal force is not a true forceexerted on your body. • It is simply your tendency to move in a straight line due to inertia.

  5. Quick review • Tangents to circles 3 THM • Arcs and chords 1 P; 4 THM • Inscribed Angles 4 THM

  6. Other Angle Relationships in Circles NCSCOS: 2.02; 2.03

  7. Essential Question • How do we use angles formed by tangents and chords to solve problems in geometry? • How do we use angles formed by lines that intersect a circle to solve real life problems?

  8. Using Tangents and Chords • You know that measure of an angle inscribed in a circle is half the measure of its intercepted arc. This is true even if one side of the angle is tangent to the circle. mADB = ½m

  9. Drawing activity. • Draw a circle. Then draw a chord. • Draw a tangent line that intersects the circle at one of the points that the chord intersects the circle at as well. • Measure the angles formed between the tangent line and the chord. • Then measure of the intecepted arcs formed. • What is the relationship between each angle and it’s intercepted arc? • Conjecture: If a tangent and a chord intersect at a point on a circle then the measure of each angle formed is_____the measure of the intercepted arc.

  10. Theorem 10.12 • If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. m1= ½m m2= ½m

  11. Finding Angle and Arc Measures • Line m is tangent to the circle. Find the measure of the red angle or arc. • Solution: m1= ½ m1= ½ (150°) m1= 75° 150°

  12. Finding Angle and Arc Measures • Line m is tangent to the circle. Find the measure of the red angle or arc. • Solution: m = 2(130°) m = 260° 130° m

  13. Finding an Angle Measure • In the diagram below, is tangent to the circle. Find mCBD • Solution: mCBD = ½ m 5x = ½(9x + 20) 10x = 9x +20 x = 20 mCBD = 5(20°) = 100° (9x + 20)° 5x° D

  14. Lines Intersecting Inside or Outside a Circle • If two lines intersect a circle, there are three (3) places where the lines can intersect. on the circle

  15. Inside the circle

  16. Outside the circle

  17. Lines Intersecting • You know how to find angle and arc measures when lines intersect ON THE CIRCLE. • You can use the following theorems to find the measures when the lines intersect INSIDE or OUTSIDE the circle. • Let’s work with the drawing activity first

  18. Drawing activity. • Draw two chords that intersect inside the circle. • Measure one of the angles created by the two chords. Find the sum of the 2 intercepted arcs formed by the 1 angle formed by the chord and its vertical angle. • What is the relationship between those two items. • Conjecture: If two chords intersect in the interior of a circle, then the measure of each angle is _______of the measures of the arcs intercepted by the angle and its vertical angle.

  19. Drawing activity • Draw a circle. Pick an exterior point. • Draw a secant that goes through the exterior point. Draw a tangent that goes through the exterior point. • Measure the angle formed. • Measure the intercepted arcs & find their difference. • Conjecture: If a tangent and a secant intersect in the exterior of a circle, then, the measure of the angle formed is _________the difference of the intercepted arcs.

  20. Drawing activity • Draw a circle. Pick an exterior point. • Draw 2 secants that intersect at this exterior point. • Measure the angle formed. • Measure the angles of the intercepted arcs, see their difference. • Conjecture: If 2 secants intersect in the exterior of a circle, then, the measure of the angle formed is _________the difference of the intercepted arcs.

  21. Drawing activity • Draw a circle. Pick an exterior point. • Draw 2 tangents that intersect at this exterior point. • Measure the angle formed. • Measure the angles of the intercepted arcs and find their difference. • Conjecture: If a 2 tangents intersect in the exterior of a circle, then, the measure of the angle formed is _________the difference of the intercepted arcs.

  22. m1 = ½ (m + m ) m2 = ½ (m + m Theorem 10.13 • If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

  23. m1 = ½ (m - m ) Theorem 10.14 • If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

  24. Theorem 10.14 • If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. m2 = ½ ( m - m )

  25. Theorem 10.14 • If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 3 m3 = ½ (m - m )

  26. Finding the Measure of an Angle Formed by Two Chords 106° • Find the value of x • Solution: x° = ½ (m +m x° = ½ (106° + 174°) x = 140 x° 174° Apply Theorem 10.13 Substitute values Simplify

  27. mGHF = ½ m( - m ) Using Theorem 10.14 200° • Find the value of x Solution: 72° = ½ (200° - x°) 144 = 200 - x° - 56 = -x 56 = x x° 72° Apply Theorem 10.14 Substitute values. Multiply each side by 2. Subtract 200 from both sides. Divide by -1 to eliminate negatives.

  28. mGHF = ½ (m - m ) Using Theorem 10.14 Because and make a whole circle, m =360°-92°=268° x° 92° • Find the value of x Solution: = ½ (268 - 92) = ½ (176) = 88 Apply Theorem 10.14 Substitute values. Subtract Multiply

  29. Central Angle Vertex OUTSIDE circle Vertex ON circle Vertex INSIDE circle Arc Angle Formulas

  30. Case I: Vertex is ON the circle ARC ANGLE ANGLE ARC

  31. 224° Find m1. 1 m1 = 68

  32. Case II: Vertex is inside the circle A ARC B ANGLE D ARC C Looks like a PLUS sign!

  33. Find m1. 92° A B 1 D C 126° m1 = 109

  34. Case III: Vertex is outside the circle C ANGLE small ARC A LARGE ARC D B LARGE ARC LARGE ARC small ARC ANGLE small ARC ANGLE

  35. Find m1. 1 15° A D 65° B m1 = 25

  36. Find mAB. mAB = 16 A 27° 70° B

  37. Find m1. 260° 1 m1 = 80

  38. In the company logo shown, =108°, and = 12°. What is mFKH? Design Application

  39. Describing the View from Mount Rainier • You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level. Find the measure of the arc that represents the part of Earth you can see.

  40. Describing the View from Mount Rainier • You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level. Find the measure of the arc that represents the part of Earth you can see.

  41. Describing the View from Mount Rainier • and are tangent to the Earth. You can solve right ∆BCA to see that mCBA  87.9°. So, mCBD  175.8°. Let m = x° using Trig Ratios

  42. 175.8  ½[(360 – x) – x] 175.8  ½(360 – 2x) 175.8  180 – x x  4.2 Apply Theorem 10.14. Simplify. Distributive Property. Solve for x. From the peak, you can see an arc about 4°.

  43. Lesson Quiz: Part I Find each measure. 1. mFGJ 2. mHJK 41.5° 65°

  44. Lesson Quiz: Part II 3. An observer watches people riding a Ferris wheel that has 12 equally spaced cars. Find x. 30°

  45. Lesson Quiz: Part III 4. Find mCE. 12°

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