# Warm Up #23: Review of Circles 1.) A central angle of a circle is an angle with its vertex at the of the circle. Example:

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1 Geometr hapter 12 Notes Warm Up #23: Review of ircles 1.) central angle of a circle is an angle with its verte at the of the circle. Eample: X 80 2.) n arc is a section of a circle. Eamples:, 3.) We measure arcs in Eample: m 4.) minor arc is a section of a circle that is less than. Eample of a minor arc: 5.) major arc is a section of a circle that is greater than. Eample of a major arc: 6.) The entire arc of a circle measures 7.) semicircle is of a circle (formed b a ) and measures. 8.) Eamples of a semicircle:, E The measure of an arc is equal to the measure of its angle. rc ddition Postulate The sum of the measures of two adjacent arcs is equal to the measurement of the larger arc that the form together. 50 G 80 m m m F 40 Find m<1: 9.) 10.) 11.)

2 - 2 - Notes #23: Tangents (Sections 12.1) Tangents 1.) is a segment, ra, or line that touches a circle onl once. Name four tangents:,, 2.) The point where a tangent touches a circle is called the. X Y Z 3.) Name the point of tangenc: Theorem 12.1 tangent is to the drawn to the point of tangenc. raw this relationship to the left. Use this theorem to complete: K For #4-5, assume that the lines that appear to be tangent are tangent. J T For #6-8, JT is tangent to the circle at T 6.) If T = 5, JT = 10, J = 4.) If = 100, find m 7.) If m J 30 and J = 12, JT = 5.) If m 40, find 8.) If JK = 18 and K = 7, then JT =

3 - 3 - Theorem 12-3 Tangents to a circle from a common point are. Use this corollar to complete: 9.) X = 5, X = X 10.) Find the perimeter of 10 cm F 8cm 15 cm E 11.) Find the perimeter of the quadrilateral: 1.3 in 5.7 in 3.4 in 3.6 in 12.) We sa that a polgon is a circle when all vertices (corners) are on the circle. In this case, we can also sa that the circle is about the polgon. 13.) escribe this figure in two was: 14.) escribe this figure in two was: G P Q F H 15.) raw a circle inscribed in a square. 16.) raw a circle circumscribed about a pentagon.

4 - 4 - ompass Practice: (When using our compass, ou must keep the point (called the center) still.) 17.) Use our compass to make small 18.) Use our compass to make a large circle. circles. Label their centers. Label the center. 19.) Use our compass to draw arcs of a circle. onstruction #1: Given a segment, construct a segment congruent to the given segment (pg. 44) Steps: 1.) Using our straightedge, draw a segment longer than. Pick a point on this segment and label it X. [ur goal is to measure and mark (or cut and paste) onto our new segment.] 2.) Set the width (or radius) of our compass to the length of. This means place the point of our compass on, and then stretch our compass so that the pencil is on point. [You have now stored this length into our compass.] 3.) Without changing the opening of our compass, place its point on X. Use the compass to mark our segment with an arc. Label this intersection as Y. 4.) heck with a ruler that XY onstruct a segment congruent to each given segment: 20.) 21.)

5 Warm Up #24: 1.) Find the perimeter of the quadrilateral: ) is tangent to circle at. If = 12 and m 30, find the length of the radius and the diameter of the circle ) Find the area of a regular heagon with perimeter 96m. 4.) Find the volume of a clinder with diameter 8in and height 10in. Notes #24: Section 12.2 and onstructing ongruent ngles hords (Section 12.2) 1.) and MN are. These segments connect an points on a circle. 2.) What is a name for the longest chord in a circle? 3.) and MN are. These are lines that contain a. M N Theorem 12.4 Y (1) ongruent arcs have congruent. (2) ongruent chords have congruent. X

6 - 6-4.) m 5.) m 150 Theorem 12.6 diameter or radius that is to a chord bisects the and its. Use this theorem to complete: (look for right triangles!!) 6.) XY =, iameter = Y M X ) PQ = 16, M =, iameter = 10 Q M P Theorem 12.5 (1) hords that are from the center are (2) chords are from the center **remember that distance is measured with a perpendicular segment, so look for right angles and was to make right triangles** Use this theorem to complete: 8.) M = N = 6, M = 8, EF =, radius =, diameter = E M N F

7 - 7 - onstruction #2: onstruct an angle congruent to the given angle. (pg. 45) Steps: 1.) Using our straightedge, draw a segment. (This will be the base edge of our angle.) Label the left endpoint of the segment E. (This will be the verte of our new angle.) 2.) Set the width (or radius) of our compass to a length shorter than. 3.) Place the center (or point) of our compass at point and draw an arc that passes through both sides of the original angle. Mark these two intersection points as X and Y. 4.) Without changing the width of our compass, draw this same arc with the compass point at E. Mark the intersection point on the base as M. 5.) Re-set our compass to the distance between intersection points X and Y. (This measures the width of the original angle.) 6.) Without changing the width of our compass, put the point of our compass on M and draw an arc. Mark this intersection as. 7.) onnect points and E. heck with our protractor that EF onstruct an angle congruent to each given angle: 9.) 10.)

8 - 8 - Warm Up #25: 1.) T = 6, RS = 8, R =, radius =, diameter = S R T 2.) =12, M=8, N=4, EF =, radius =, diameter = M F E N 3.) Solve for : 4 20 Notes #25: Spheres, Inscribed ngles, onstructing Perpendiculars (Section 12.3) plane passes h cm from the center of a sphere with radius R, forming a circle with radius r. Find the indicated values: 1.) R = 5, h = 4, r =? 2.) r = 9, h = 12, R =?

9 n inscribed angle is an angle whose verte is the circle. Eample: 1 2 The measure of an inscribed angle equals the measure of the intercepted arc. Eample: m 1=, m 2 = Special ases: n inscribed angle that intercepts a semicircle is Eample: X The measure of an angle formed b a tangent and a chord equals the measure of its intercepted arc. Eample: Y 120 Z 1 If two inscribed angles intercept the same arc, then the angles are. Eample: 1 2 F G If a quadrilateral is inscribed in a circle, then its opposite angles are E Eample: H Solve for the indicated variables: 3.) =, = 4.) =, = 60 20

10 ) =, = 6.) =, = ) =, =, z = 95 z 8.) =, =, z = z onstructing Perpendicular Lines onstruction #3: Given a point on a line, construct the perpendicular to the line at the given point (pg. 182) 1.) Given point on line k 2.) Using as our center, draw two arcs on line k, one arc to the left of and one to the right of. 3.) Label these new points as X and Y. 4.) Place the center of our compass on X and using a radius larger than X, draw one arc above line k and one arc below line k. 5.) o the same from point Y. oth pairs of arcs must intersect above and below line k. Label these points M and N. 5.) onnect M and N. This segment should intersect point and be perpendicular to line k.

11 onstruct a perpendicular through the indicated point: 9.) 10.) onstruction #4: Given a point outside a line, construct the perpendicular to the line from the given point (pg. 183) Steps: 1.) Place the center of our compass on point P. 2.) Using a consistent width, draw two arcs that intersect line k. Label these points X and Y. 3.) From X, draw an arc below line k 4.) From Y, draw an arc below line k 5.) These two arcs must intersect. Label this intersection Z. 6.) onnect P and Z; this segment is perpendicular to line k. onstruct the perpendicular segment from P through line k. 11.) 12.) P P k k

12 Warm Up #26: Quiz Review 1.) Find the perimeter of the quadrilateral ) is tangent to circle at. If = 18 and m 30, find the length of the radius and the diameter of the circle ) Name one of each: a) central angle b) chord c) secant d) inscribed angle X Y 4.) T = 12, ST = 16, R =, radius =, diameter = T S R 5.) z w = = = w 80 z =

13 Warm Up #27: Solve for the variables ) 2.) z ) 50 z v 20 w 4.) =12, M=3, N= 2 5, EF =, radius =, diameter = M E N F Notes #27: ngle Measures and Segment Lengths (Section 12.4), Review of onstructions Interior and Eterior ngles Theorem (Part I) a b n angle formed b two chords is equal to the of its two intercepted arcs. (interior angle) = (arc + arc) 2 Solve for : 1.) 2.) 3.)

14 Theorem (Part II) n eterior angle formed b (a) two secants, (b) two tangents, or (c) one tangent and one secant is equal to the of its two intercepted arcs. (eterior angle) = (arc - arc) 2 (a) (b) (c) a b a b a b Solve for and : 4.) 5.) ) 7.)

15 omplete the constructions: 8.) onstruct a segment congruent to the given segment: 9.) onstruct an angle congruent to the given angle: M L 10.) onstruct a line perpendicular to the given line through the given point: 11.) onstruct a line perpendicular to the given line through the given point: 12.) onstruct a line perpendicular to the given line through the given point: 13.) onstruct a line perpendicular to the given line through the given point: 14.) onstruct an angle congruent to the given angle:

16 Warm Up #28: ircle Review 1.) Find M and MQ 2.) Find M and MQ P M Q P M Q 3.) Find and 4.) Find and Notes #28: Segment Lengths, onstructing Parallel Lines and Perpendicular isectors (Section 12.4) Segment Lengths in ircles Theorem (Part I) a b When two chords intersect in a circle, the product of the of one chord equals the product of the of the other chord (piece)(piece) = (piece)(piece) Solve for : 1.) 2.)

17 Theorem (Part II) When (a) two secants or (b) one tangent and one secant are drawn to a circle, then: (a) (eterior segment)(whole length) = (eterior segment)(whole length) (b) a b a b Solve for the variables: 3.) 4.) ) 6.) ) 6 8.) )

18 onstruction #5: Given a point outside a line, construct the parallel to the given line through the given point (pg. 181) Steps: 1.) raw points and, in that order and not too far apart, on line k. 2.) raw P and etend this ra a considerable distance. (t P, we will construct an angle congruent and corresponding to P.) 3.) Set our compass to the width of. With the center of our compass at, draw an arc that passes through line k and P. Mark the new point of intersection as X. 4.) Without changing the width of our compass, put its center at P and draw a similar arc above and to the right of point P. Mark the intersect as M. point where this arc and P 5.) hange the width of our compass to be the length X. 6.) Keeping this width, put the center of our compass on M and draw an arc that intersects the previous arc. Mark this point of intersection as N. 7.) raw PN. This line is parallel to line k. onstruct the parallel line to k through P. 10.) 11.) P P k k onstruction #6: onstruct a perpendicular bisector to a given segment (pg. 46) Steps: 1.) Set the width (or radius) of our compass to a length longer than half the given segment. 2.) With the center of our compass on point, draw one arc above and one arc below the segment. 3.) With the center of our compass on point, draw one arc above and one arc below the segment. 4.) Make sure that ou find where these pairs of arcs intersect. Mark these points as X and Y. 5.) onnect X and Y; this is our perpendicular bisector.

19 ) 13.) Warm Up #29: ircle Review Solve for the variables: 1.) 2.) ) 4.) Notes #29: ircles in the oordinate Plane, onstructing ngle isectors (Section 12.5) ircles Equations of circles are written in this form: ( h) ( k) r Where (h, k) is the of the circle and r is the of the circle Graph and find the equation of each circle: Graph the center point From this point, go the distance of the radius up, down, left, and right onnect these 4 points as a circle

20 1.) (-2, 4) r = 4 2.) (5, 0), r = 3 Name the center and radius of each circle, then graph ) 4.) 2 2 ( 1) ( 2) 9 ( 4) Write the standard equation of each circle: Find the center point From this point, count the distance to the highest point on the circle. This is the radius. 5.) 6.)

21 Write the standard equation of the circle with the given center that passes through the given point: Plug the center point into the standard equation as (h, k) Plug the second point into that equation as (, ) Solve for r. Plug into equation from first step. 7.) circle has a center (1, 3) and passes through the point (4, -1). Find the equation of this circle and graph it. 8.) Find the equation of the circle whose center is (-3, -2) and goes through the point (-1, 2). onstruction #7: onstruct the bisector of a given angle (pg. 47) Steps: 1.) Set the width (or radius) of our compass to a length shorter than. 2.) Place the center (or point) of our compass at point and draw an arc that passes through both sides of the original angle. Mark these two intersection points as X and Y. 3.) Keeping the same width, place the center of our compass at point X and draw an arc in the interior of the angle. o the eact same process with our compass at point Y. 4.) Label the intersections of these two interior arcs Z 5.) Using our straightedge, draw Z 6.) Z bisects Using onl a compass and a straightedge, bisect the given angles: 9.) 10.)

22 HW #30 Geometr hapter 12 Stud Guide Name : Necessar Skills: Pthagorean Theorem and Special Right Triangles. Solve for and 1.) 2.) 3.) 4.) Ke Vocabular: Name one eample of each 5.) a. radius b. diameter E c. chord d. secant e. tangent f. point of tangenc g. central angle i. minor arc h. inscribed angle j. major arc 6.) raw each: b. circle circumscribed about an obtuse triangle. a. rectangle inscribed in a circle Tangents: 7.) omplete: JT is tangent to the circle at T a.) If T = 4, J = 12, then JT = b.) If m J 30 and JT= 12, J =, T = c.) If JK = 4 and K = 6, then JT = K J T entral and Inscribed ngles: Solve for and 8.) 9.) ) ) ) ) F G 110 E H

23 ) ) ) hords and rcs: 17.) ) T = 12, RS = 4, R =, radius =, diameter= S R T 19.) = 6, M = 4, N = 3, EF =, radius =, diam. = M F E N 20.) EF = 8, N = 3, M = 2, = M F E N Interior and Eterior ngles: Solve for 21.) 22.) 23.) ) 25.) 26.) Segments and Lengths: Solve for 27.) 28.) 29.)

24 ) 31.) 6 32.) Find the perimeter of each polgon: 33.) 34.) 7 cm 13 cm 1.9 in 3.7 in 3.4 in 5 cm 6 cm 3.6 in omplete the problems about circles: 35.) Find the equation of the circle with center at (2, -7) and radius ) Find the equation of the circle whose center is (-3, -1) and goes through the point (1, 2) 37.) Find the center and the radius of the circle described b: ( - 5) 2 + ( +3) 2 = 9. Graph ) chord is 4m from the center of a circle. If the radius of the circle is 10m, how long is the chord? 38.) Find the equation of the circle shown here: 40.) sphere is sliced b a plane 2ft from the center of the sphere. If the radius of the circular plane made b the plane s slice is 8ft, what is the radius of the sphere? **on t forget worksheet problems 40-48**

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