Intro to Circles Formulas Area: Circumference: Circle:

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1 Intro to ircles Formulas rea: ircumference: ircle: Key oncepts ll radii are congruent If radii or diameter of 2 circles are congruent, then circles are congruent. Points with respect to ircle Interior Points:. Exterior Points:.. On ircle: Identify Parts of ircle Given the circle to the right, identify the following: 1. Name the circle 2. Name the radii E 3. Name the chords 4. Name a diameter Example: Finding Radius and iameter 1. ircle has diameters MO and QN a. If QN = 10, find Q 10 = 2(Q) 5 = Q b. If M = 7, find MO MO = 7(2) MO = 14 c. If P = 12, find O O = 12 since all radii are congruent 2. ircle H has a diameter of 30, and circle K has a diameter of 20 units. If JI = 6, find HI. HJ = ½(30) = 15 HJ JI = HI so 15 6 = 9 1 Rev

2 Practice: Finding Radius and iameter 3. ircle has diameters MO and QN a. If QN = 18, find Q b. If M = 24, find MO c. If P = 2, find O 4. ircle H has a radius of 20 units, and circle K has a radius of 16 units. If JI = 6, find HI. Example: Finding ircumference, diameter or radius 1. Find when 2. Given = 136.9m find a. r = 7 = 2π(7) = 14π b. d = 12.5 = 12.5π a. d = dπ = d b. r = 2πr or r = ½(d) = = r 2 Practice: Finding ircumference, diameter or radius 3. Find when 4. Given = 65.4m find a. r = 13 b. d = 6 a. d b. r Example: Use other figures to find circumference Find the circumference of circle P. iameter of circle is hypotenuse of right triangle = c 2 c = 13 5 = 13π 12 Practice: Use other figures to find circumference Find the circumference of circle P Rev

3 Equation of ircle Equation (x-h) 2 + (y-k) 2 = r 2 where (h,k) is the center of the circle and r is the radius Key oncepts In order to write an equation of a circle or to graph a circle, you need the center and radius of the circle. on t forget that the diameter is twice the radius. Examples: Writing Equations of ircle Write an equation for each circle. 1. enter at (-2, 5), d = 4 2. enter at origin, r = 3 3. enter at (-12, -1), r = 8 Step 1: Identify h & k h = -2, k = 5 Step 2: Identify r d = 4 r = 4/2 = 2 Step 3: Substitute in (x-(-2)) 2 + (y-5) 2 = 2 2 (x+ 2) 2 + (y-5) 2 = 4 Step 1: Identify h & k h = 0, k = 0 Step 2: Identify r r = 3 Step 3: Substitute in (x - 0) 2 + (y-0) 2 = 3 2 x 2 + y 2 = 9 Step 1: Identify h & k h = -12, k = -1 Step 2: Identify r r = 8 Step 3: Substitute in (x (-12)) 2 + (y-(-1)) 2 = 8 2 (x+ 12) 2 + (y+1) 2 = 64 Practice: Writing Equations of ircle Write an equation for each circle. 4. enter at (-2, 5), d = enter at origin, r = 7 6. iameter with endpoints at (2, 7) and (-6, 15) Examples: Graphing Equations of ircle Graph the equations 7. (x+ 2) 2 + (y-3) 2 = 16 Step 1: Identify the center h = -2, y = 3; enter (-2, 3) Step 2: Identify the radius r = 16 = y Step 3: Plot the center. Use a compass set at a width of 4 grid squares to draw the circle. 8. x 2 + y 2 = x Rev

4 Key oncepts ngles and rcs Sum of measure of central angles of circle with no interior points in common is 360 rc: - Measure: - length: Theorems/Postulates: ongruent Intercepted rcs: If 2 central angles of circle (or of congruent circles) are congruent, then their intercepted arcs are congruent. ongruent entral ngles: If 2 arcs of circle (or of congruent circles) are congruent, then corresponding central angles are congruent. rc ddition Postulate: measure of arc formed by 2 adjacent arcs is the sum of measures of the 2 arcs. Examples: Measures of entral ngles 1. In circle, m E = 49, m = 59, Find m. 2. In circle, m E = 49, m = 59, Find m E. Practice: Measures of entral ngles RV is diameter of circle T. 1. Find m RTS R Q S T 8x-4 2. Find m QTR 20x (13x-3) (5x+5) V U Examples: Identify rcs Given ST is the diameter of ircle O, identify the following: 1. Semicircle: 2. Major rc: 3. Minor rc: 4 Rev

5 Examples: Measures of rcs In circle F, m F = 50 and F F. Find each measure a. m = FE F 50 b. m + = = 140 c. m + = = 230 Note: is a semicircle Practice: Measures of rcs In circle P, m NPM = 46 and bisects KPM and. Find each measure d. m e. m f. m Examples: rc Lengths 1. In circle P, = 15 and QPR = 120. Find the length of. Step 1: Find the circumference of circle P = 2πr = 2π(15) = 30π Step 2: find the degree measure of. egree measure of rc = Step 3: Multiply results from step 1 and step 2 together. 1 * 30π = 10π 3 2. Using the same circle above, find the length of when PR = 4.5 and QPR = Q P R Practice: rc Lengths iameter of circle is Find the length of if m ON = Find the length of if m QM = 90 5 Rev

6 rcs and hords hord/rc/entral ngle Theorems In circle or circles, 2 minor arcs are if and only if their corresponding chords are. In circle or circles, 2 chords are if and only if their corresponding minor arcs are. If 2 central angles of circle (or of circle) are, then corresponding chords are. If 2 chords of circle (or of circles) are, then corresponding central angles are. If 2 arcs of circle (or of circles) are, then the corresponding chords are. If 2 chords of circle (or of circles) are, then the corresponding arcs are. Summary: In same or circles, chords arcs central s. Radius/iameter-hord Theorems The distance from center of circle to chord is measure of perpendicular segment from center to chord. If radius/diameter is perpendicular to a chord, then it bisects chord. Example: If, then and T U V If radius of circle bisects chord that isn t the diameter, it is perpendicular to that chord. Perpendicular bisector of chord passes through center of circle. In circle or in circles, 2 chords are if and only if they are equidistant from the center. If, then E If, then O F Examples: Radius-hord Relationships 1. ircle O has a radius of 13in. Radius is perpendicular to chord, which is 24in long. a. if m = 134, find m bisects so = ½m = ½(134) = 67 b. Find OX O = 13 O bisects X = ½() = ½(24) = 12 Use Pythagorean Theorem to find OX (OX) 2 = (O) 2 (X) 2 OX = 5 2. In circle Q, UV = 8x + 1and RT = 9x 4. Find x. 6 Rev

7 3. hords and F are equidistant from the center. G = 10. If radius of circle G is 26, find & E. Step 1: Since and F are equidistant, F F Step 2: raw G and GF to form right triangles. Step 3: Use Pythagorean Theorem to solve. E G () 2 = (G) 2 (G) 2 = = 576 = 24 Step 4: Solve for = ½(); therefore, = 2(24) = 48 Step 4: Solve for E F; therefore, E = ½(); therefore, E = ½(48) = 24 Practice: Radius-hord Relationships 4. ircle O has a radius of 12in. Radius O is perpendicular to chord, which is 12in long. a. if m = 150, find m b. Find OX 5. ircle O has a radius of 10in. Radius O is perpendicular to chord, which is 10in long. a. if m = 60, find m b. Find OX c. Find X 6. hords and F are equidistant from center. If radius of circle G is 15 & = 24, find G & E. F E G 7 Rev

8 7. FH FL, FK = 17 and FH = 8. Find LK, KM and JG. J K H F L G M Inscribed ngles Inscribed ngle Theorems Inscribed ngle Theorem: measure of inscribed angle = ½ measure of intercepted arc ongruent Inscribed ngle Theorem: Inscribed of arcs or same arc are Right ngle Inscribed Theorem: If inscribed intercepts a semicircle, then the is a right Opposite ngles of Inscribed Quad: if a quadrilateral is inscribed in circle, then its opposite s are supplementary. Examples/Practice 1. Given circle O, find the measure of when arc = 60..O 2. For circle N, find the measure of the following: a. m KJL = ½ ( ) = ½ (60) = 30 b. m KLJ c. m JKL 3. Find the value of x. 4. Find the value of x. 8 Rev

9 5. and are inscribed in circle F with. Find the measure of,, and if = 12x 8 and = 3x Quadrilateral is inscribed in circle P. = 80, = 40. Find and. 7. Find each measure a. EGF = b. EG = 8. = 104 and = 67. Find and. c. GE = d. GFE = e. GEF = 9. arc = 110 and arc PQ = 100. Find R and P. 10. Find JKL if arc JLM = 180. P Q S R 9 Rev

10 Tangent Theorems Tangents If a line is tangent to circle, then it is perpendicular to the radius drawn to the point of tangency. R T O If RT is a tangent, OR RT If a line is perpendicular to a radius of circle at its endpoint on the circle, then the line is tangent to the circle. R T O If OR RT, then RT is a tangent. Two-Tangent Theorem: If 2 segments from same exterior point are tangent to circle, then those segments are congruent. 4 E Examples: Find lengths 3 1. E is tangent to circle F at point E. Find x. x F Examples: Identify Tangents 4 M N 2. NO = 2. etermine whether MN is tangent to circle L. O 3 L P 5 Q 3. PS = 4. etermine whether PQ is tangent to circle R. 4 S 4. = -2x + 37 and = 6x + 5. Find x. ssume that segments that appear tangent to circles are tangent. 10 Rev

11 Practice: Tangents 5. E is tangent to circle F at point E. Find the diameter of the circle. 6. = 7, = 7. etermine whether is tangent to circle. 7. GE = y 5, E = 10, F = y and FH = x + 4. Find x. ssume that segments that appear to be tangent to circles are tangent. G E H F 8. is circumscribed about circle O. F = 6, = 19 and E = E + F. Find the perimeter of. E F 9. HJK is circumscribed about circle G. Find perimeter of HJK if NK = JL + 29, LH = 18 and JM = 16 K M N J L H 11 Rev

12 Theorems/Postulates Secants, Tangents and ngle Measures The measure of an inscribed angle or a tangent-chord angle (vertex on circle) = ½ measure of intercepted arc. = ½ EF = ½ E F The measure of a chord-chord angle = ½ sum of the intercepted arcs G K GHK = ½ ( + ) H GHI = ½ ( + ) I J The measure of secant-secant, tangent-secant and tangent-tangent angle = ½ difference of intercepted arcs. V T N M L W U S O P R Q Summary X Secant- Secant Tangent-Tangent Tangent-Secant LNP = ½ ( - ) UVW = ½ ( - ) QST = ½ ( - ) Vertex on enter: angle = intercepted arc Vertex on ircle: angle = ½ (intercepted arc) Vertex inside circle: angle = ½ (arc1 + arc2) = average of bases Vertex outside circle: angle = ½ (arc1 arc2) = difference of arcs Examples Find the measures of the missing angles or arcs. 1. = 104, = 20 & is 2. = 233, = 127 diameter Find UVW Find LNP 3. QST = 32, = 125 Find 12 Rev

13 4. = 65. Find 5. = 27 & = 88. Find GHK 6. = 258. Find EF Practice Find the measures of the missing angles or arcs. 7. = 57, = 31 & is diameter Find LNP 8. = 233, UVW= 53 Find 9. QST = 55, = 40, = 6x. Find x 10. = 100. Find 11. = 29 & = 47. Find GHK 12. EF = 102. Find 13 Rev

14 Theorems/Postulates Special Segments in ircle hord-hord Power Theorem: If 2 chords of circle intersect inside circle, then the product of measures of segments of the chords are equal. E * E = E * E E Tangent-Secant Power Theorem: If tangent segment & secant segment are drawn from external point to circle, then square of measure of tangent segment equals product of entire secant segment and its external part. R (TP) 2 = (PR)(PQ) Q T P Secant-Secant Power Theorem: If 2 secant segments are drawn from external point to circle, then product of measures of 1 secant segment & its external part is equal to product of measures of other secant segment and its external part. (product of exterior part & whole segment) P * P = P * P P Examples 1. Find x, y and z. a. b. c. 2. T is the midpoint of, = 8 and = 40. a. Find b. Find the diameter of circle O. 14 Rev

15 Practice Find the following x+1 x 7 9. T is the midpoint of, = 4 and = Find - Find the diameter of circle O. 15 Rev