1 / 49

Circles

Learn about the different elements of circles, including interior and exterior angles, chords, tangents, and their relationships. Explore calculations and examples to deepen your understanding.

sriddick
Download Presentation

Circles

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Circles

  2. Vocabulary • Interior • Exterior • Radius • Diameter • Chord • Secant • Tangent

  3. chords VT and WR secant: VT tangent: s diam.: WR radii: QW and QR Lesson Quiz: Part I 1. Identify each line or segment that intersects Q.

  4. Circles • Congruent • Concentric • Tangent • Internally tangent • Externally tangent

  5. Tangent Line Radius

  6. Tangents • Tangent • Perpendicular to radius • Example: Visible distance to horizon from Mt Everest • Mt Everest is 29,000 ft above sea level • Find: EH • 2 tangents from external point • Same measure

  7. 12-2 Arcs & Chords • Types of Arcs • Minor Arc • < 180o • Major Arc • >180o • Semi-Circle • = 180o • Central Angle: a angle whose vertex is at the center of a circle

  8. b. mAHB a. mFMC mFMC = 0.30(360) = 108 Central is 30% of the . = 360° – mAMB = 0.10(360) c. mEMD mAHB =360° –0.25(360) = 36 = 270 Central is 10% of the .

  9. Solving for Arcs: • Adjacent Arcs: • Arc Addition Postulate:

  10. mJKL mKL = 115° mJKL = mJK + mKL Find each measure. mLJN mKPL = 180° – (40 + 25)° Arc Add. Post. = 25° + 115° Substitute. = 140° Simplify.

  11. Congruent Arcs – 2 Arcs with the same measure • Central Angles • Chords

  12. Congruent Arcs • Examples • Find RT • Find mCD

  13. Radii & Chords • If radius is perpendicular to chord • Bisects Chord & Arc • A perpendicular bisector of chord is a radius

  14. Step 1 Draw radius RN. RM NP , so RM bisects NP. Using Radii and Chords Find NP. RN = 17 Radii of a are . Step 2 Use the Pythagorean Theorem. SN2 + RS2 = RN2 SN2 + 82 = 172 Substitute 8 for RS and 17 for RN. SN2 = 225 Subtract 82 from both sides. SN= 15 Take the square root of both sides. Step 3 Find NP. NP= 2(15) = 30

  15. Step 1 Draw radius PQ. PS QR , so PS bisects QR. Find QR to the nearest tenth. PQ = 20 Radii of a are . Step 2 Use the Pythagorean Theorem. TQ2 + PT2 = PQ2 TQ2 + 102 = 202 Substitute 10 for PT and 20 for PQ. TQ2 = 300 Subtract 102from both sides. TQ 17.3 Take the square root of both sides. Step 3 Find QR. QR= 2(17.3) = 34.6

  16. Examples

  17. Inscribed Angles • Inscribed angle • Vertex on circle • Sides contain chords • Measure of inscribed angle = ½ measure of arc • m<E = ½(mDF)

  18. Inscribed Angles • If inscribed angle arcs are congruent • Intercept same arc or congruent arcs • THEN: Inscribed angles are congruent

  19. Example • Find mBC • Find m<ECD • Find m<DEC

  20. Example 2: Hobby Application An art student turns in an abstract design for his art project. Find mDFA. mDFA= mDCF + mCDF = 115°

  21. Inscribed Angle • Inscribed angle subtends a semicircle if and only if the angle is a right angle • Example:

  22. Quadrilaterals • Opposite Angles Add to 180

  23. Example: Finding Angle Measures in Inscribed Quadrilaterals Find the angle measures of GHJK. Step 1 Find the value of b.

  24. Angle Relationships • Tangent and a secant/chord • Measure of angle is ½ intercepted arc measure • Measure of the arc is twice the measure of angle • Example: • Find m<BCD • Find measure of arc AB

  25. Interior Angle • Intersect inside circle • Measure of vertical angles is ½ sum of arcs • Example: • Find m<PQT

  26. Exterior Angle

  27. Examples • Find x 1. 2. 3.

  28. 4. Find mCE. 12°

  29. Arc Length/Sector/Segment

  30. Sectors & Arc LengthGeometry 12.3 • Sector of a circle – 2 radii & arc • The pie shaped slice of the circle • Area of sector is percent area of circle based on arc or central angle: • A= Area, r= radius, m= measure of arc/angle

  31. Segments of Circles • Segment of circle • Area of arc bounded by chord • Finding Area of Segment

  32. Segment Examples Segment: • segment RST - segment DEF Area sector-Area of Triangle

  33. Arc Length • Distance along the arc (circumference) • Measured in linear units • L= length, r= radius, m= measure of arc/angle • Example: • Measure of GH

  34. J Segment RelationshipsGeometry 12.6 J Example:

  35. Example: Art Application The art department is contracted to construct a wooden moon for a play. One of the artists creates a sketch of what it needs to look like by drawing a chord and its perpendicular bisector. Find the diameter of the circle used to draw the outer edge of the moon.

  36. Example 2 Continued 8(d – 8) = 9  9 8d – 64 = 81 8d = 145

  37. External Secant Relationships Example:

  38. External Tan/Sec Relationship Example:

  39. Lesson Review: Part I 1. Find the value of d and the length of each chord. d = 9 ZV = 17 WY = 18 2. Find the diameter of the plate.

  40. Lesson Review: Part II 3. Find the value of x and the length of each secant segment. x = 10 QP = 8 QR = 12 4. Find the value of a. 8

  41. Equation for CircleGeometry 12.7 & Algebra 2 12.2 • (x– h)2 + (y – k)2 = r2 • h is the x coordinate of the center point • k is the y coordinate of the center point • r is the radius • To determine if point is inside/on/outside • Plug x and y of point into circle equation • h & k are the CENTER POINT coordinates • Compare result to r2 • If < pt is inside : if > pt is outside : if = pt is on

  42. Distance Example: Consumer Application Use the map and information given in Example 3 on page 730. Which homes are within 4 miles of a restaurant located at (–1, 1)? The circle has a center (–1, 1) and radius 4. The points insides the circle will satisfy the inequality (x + 1)2 + (y – 1)2 < 42. Points B, C, D and E are within a 4-mile radius . Check Point F(–2, –3) is near the boundary. (–2 + 1)2 + (–3 – 1)2 < 42 (–1)2 + (–4)2 < 42 x Point F (–2, –3) is not inside the circle. 1 + 16 < 16

  43. Finding center & radius • Given 2 endpoints • Find center point • X coordinate is (x1+x2)/2 • Y coordinate is (y1+y2)/2 • Use center point coordinate and one end point with the distance formula to find the radius • Plug center point and radius into equation for circle

  44. Slope of Tangent • Slope of radius = Rise over Run (ΔY ÷ ΔX) • Find negative reciprocal • Change sign, flip fraction • Insert negative reciprocal into slope formula • Y = mx + b • Substitute y & x coords from tangent point to find b • Rewrite equation with y & x and the b value

  45. From the equation x2 + y2 = 29, the circle has center of (0, 0) and radius r = . Example: Writing the Equation of a Tangent Write the equation of the line tangent to the circle x2 + y2 = 29 at the point (2, 5). Step 1 Identify the center and radius of the circle.

  46. 2 – 5 5 2 The slope of the radius is . Because the slopes of perpendicular lines are negative reciprocals, the slope of the tangent is . Tangent Example Continued Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent. Use the slope formula. Substitute (2, 5) for (x2 , y2 ) and (0, 0) for (x1 , y1 ).

  47. Step 3 Find the slope-intercept equation of the tangent by using the point (2, 5) and the slope m = . 2 – 5 Substitute (2, 5) (x1 , y1 ) and – for m. 2 5 Tangent Example Continued Use the point-slope formula. Rewrite in slope-intercept form.

  48. The equation of the line that is tangent to x2 + y2 = 29 at (2, 5) is Tangent Example Continued Check Graph the circle and the line.

  49. Examples with graph paper

More Related