10r name: circles – day 1 central angles
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10R Name: ___________________ Circles – Day 1 Central Angles
Review Notes: Circle: A set of points equidistant from a fixed point called the center. Radius: (plural radii): A segment from the center of the circle to any point on the circle.
Theorem: All radii are
Diameter: A segment from one side of the circle to the other side of a circle through the center.
Diameter = 2(radius) Interior & Exterior Points:
D: is exterior (outside the circle)
E: is interior (inside the circle
New Vocabulary Central Angle: An angle whose vertex is the center of the circle BOA is a central angle Minor Arc: An arc less than _______________________
Written with words: minor arc AB or with symbols: AB Semi-circle: An angle = ______________________ Major Arc: An arc is greater than __________________________
Written with words: major arc BCA or with symbols: BCA
***Usually 3 letters indicates direction Intercepted Arc: Each endpoint of the arc lies on different rays of the angle.
is intercepted by AOBAB Quadrant: an arc that is one-fourth of a circle
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Degree Measure of an Arc Arc Addition Postulate
1) Equals the central angle + = AB BC ABC
2) Major Arc = 360 - minor arc + m = mmAB BC ABC 3) Semicircle = 180
In the figure, and PR QS are the diameters of .U Find the measure of the indicated arc.
9. mPQ 10. mST
11. mTPS 12. mRT
13. mRQS 14. mQR 15. 15. mPQS
16. mTQR 17. mPS 18 mPTR .
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Two diameters of T are PQ and RS . Find the given arc measure if 35mPR
24. mPS 25. mPSR 26. mPRQ 27. mPRS
Two diameters of N are JK and LM . Find the given arc measure if 165mJM
28. mJL 29. mJMK 30. mJLM 31. mKLM
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Arcs are congruent if the circles have the same central angles AND the same length radius In Exercises 39 - 40, use the following information. Sprinkler A water sprinkler covers the area shown in the figure. It moves through the covered area at a rate of about 5 per second. 39. What is the measure of the arc covered by the sprinkler? 40. If the sprinkler starts at the far left position, how long will it take for the sprinkler to reach the far right position?
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10R Geometry- Circles HOMEWORK Day 1: Central Angles Given the following diagram and diameter EB, determine whether each of the following arcs are major, minor, or semicircles.
1. ABC 2. ADC
3. BE 4. DC Find the value of x in each of the following diagrams.
5. 6. 7.
AC is a diameter. AD is a diameter.
8. 9. 10. Given the diagram to the right, with diameter AC, find each of the following values.
a. x b. BC c. BDC
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Two diameters of T are AB and CD . Find the given arc measure if 40mBC
11. mAC 12. mCAB 13. mADC 14. mDBC In each of the following problems the measure of arc RT is 90 degrees. Find the value of x in simplest radical form. 15. 16.
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10R Circles – Day 2 Inscribed Angles
Warm-up 1. If 2 sides of a triangle are 6 and 8 what is the range for the third side?
2. How many integral possibilities of the third side are there? Inscribed Angle =_________________________________________________________________ An angle whose VERTEX is ON the circle, sides are chords. Corollary: (Theorem based on another theorem)
1) __________________________________ 2)________________________________
__________________________________ _________________________________
__________________________________ _________________________________
Chord: A segment that connects two points on a circle Diameter: A chord that crosses through the center of the circle.
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Practice 1. In the figure shown, which statement is true? A) SPR PSQ B) RQS RPS C) RPS PRQ D) PRQ SQR
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10R Geometry- Circles HOMEWORK Day 2: Inscribed Angles Find the measure of the indicated angle or arc in the circles shown below.
1. m<ABC 2. mAB 3. m CSB
4. m ACD 5. mAC 6. m ASC Solve for each variable. 7. 8. 9. 10. 11. 12.
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10R Circles – Day 3 Angles made by Chords, Tangents and Secants
Warm-up: In circle O, MON is a diameter, OP is a radius and mPOM = 140. Find:
________________________ a) PM
________________________ b) PN
________________________ c) mPON
________________________ d) NMP
Chord A line that connects _______ points on a circle. This includes a diameter Tangent and Chord _________________________ Two Secants _________________________
Tangent A line that intersects the circle in exactly _______ point. Two parallel Chords _________________________ Tangent and Secant _________________________
Secant A line that intersects the circle in ________ points (crashes through 2 points) Two Chords ________________________ Two Tangents ________________________
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Practice:
A) / / , 3 2, 10,AD BC AB x CD x solve for x, find each arc.
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10R Geometry- Circles HOMEWORK Day 3: Angles made by Chords, Tangents and Secants
Solve each of the following for x. 1. 2. 3. 4. 5. 6.
7. In circle O, diameter AOB is parallel to chord CD and mAC = 48. Find the mCD
8. In circle O, chord AY is parallel to diameter DOE , chord AD is drawn, and the measure of arc AD = 40. What is the m DAY ?
9. Point P lies outside circle O, which has a diameter of AOC . The angle formed by tangent PA and secant
PBC measures 30°. Sketch the conditions given above and find the number of degrees in the measure of minor arc CB.
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10. In the accompanying diagram of circle O, diameter AOB is extended through B to external point P,
tangent PC is drawn to point C on the circle, and : 7 : 2.mAC mBC Find .m CPA
11. Given circle O with diameter GOAL; secants HUG and HTAM intersect at point H; : : 7 :3: 2;mGM mML mLT and chord GU chord UT . Find the ratio of m UGL to m H .
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10R Circles – Day 4 Angles Practice
Warm-up: Using the vocabulary, label as many parts as possible. Geometry of a Circle Circle
Radius
Diameter
Interior Point
External Point
Chord
Central Angle
Inscribed Angle
Semi Circle
Minor Arc
Major Arc
Intercepted Arc
Practice
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17. In circle O, chord AB chord AC and mBC = 60. Find: ________________________ a) mBAC
________________________ b) BA
________________________ c) BCA
18. In the accompanying diagram, AC and BD are chords
of circle O and intersect at E. If mAB = 70 and
CD = 90, find BEA
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19. In the accompanying diagram, tangent PA and secant
PBC are drawn to circle O. If mADC is twice the
measure of AB and mP = 50, what is the measure of AB ? 20. 21. 22.
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Find each of the below measures: m1: __________ m2: __________ m3: __________ m4: __________ m5: __________ m6: __________ m7: __________ m8: __________ m9: __________ m10: __________
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10R Circles – Day 4 Angles Homework
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10R Circles – Day 5 Tangents
Common Tangents: _______________________________________________________________________
________________________________________________________________________________________ Four Common Tangents: Three Common Tangents: Two Common Tangents: One Common Tangent: No Common Tangents Radius and Tangent Theorem: A radius and tangent are _____________________ at the point of tangency
Tangents drawn from the same external point are ______________________
________________________________________
Two Tangents and the line connecting the center: form an ____________________________
___________________________
___________________________
Review: _______________________________________________
_______________________________________________
_______________________________________________
_______________________________________________
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Examples
1. Find the perimeter of ABC 2. OQ = 25 and PQ = 24, find OP
3. If RT = 12 and RQ = 4, find each 4. Draw an example of a set of circles with
a) OP three tangents.
b) OQ
c) PQ
d) QS
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9. 10. 11. 12. 13. 14. 15. 16. 17.
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18. WX is tangent to Circle Y and Circle Z. What is the length of YZ? 19. Golf: A green on a golf course is in the shape of a circle. Your golf ball is 8 feet from the edge of the
green and 32 feet from a point of tangency on the green as shown in the figure. a) Assuming the green is flat, what is the radius of the green? b) How far is your golf ball from the cup at the center of the green?
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10R Geometry: Circles HOMEWORK Day 5: Tangents 1. 2. 3. 4. Kimi wants to determine the radius of a
circular pool without getting wet. She is located at point K, which is 4 feet from the pool and 12 feet from the point of tangency, as shown in the diagram. What is the radius of the pool?
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5. Draw a picture of the two circle as described: a. Intersecting but not tangent b. Internally disjoint but not concentric c. Internally tangent d. Externally tangent
6. Circle X and circle Y are tangent to each other at point P. Line l is tangent to circle X at A and to circle Y
at B. If the length of PC is 8, find the length of AB. 7. Circle O is inscribed in triangle ABC with D, E, and F the point of tangency. If AD = 6, EB = 5, and CF = 12:
a. Find the lengths of the sides of the triangle.
b. Show that the triangle is scalene. Problems 8-9: BA and BC are tangents, and DB intersects circle O at points R and D. 8. If OA = 6 and RB = 14, find OB, OA, and RB. 9. If AB=4x, CB = 6x-10, and OC = 2x + 5, find AB, CB, and OB.
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10R Circles – Day 6 Segments HW #6 page 667-668 #3-11, 16-20
Theorems for proofs: 1) central 's chords arcs 2) A diameter perpendicular to a chord: bisects the chord and its arcs 3) The perpendicular bisector of a chord passes through the center of the circle Theorem 1 Theorem 2 & 3
4) Two chords chords equidistant from the center of circle
AB CD EO FO
Pythagorean Triples:
_______________________________
_______________________________
_______________________________
_______________________________
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10R Circles – Day 7 Segments
Warm-up 1. Name the longest chord of the circle._____________________________ 2. Name the line segment that touches a circle in one point._____________________________ 3. A(n) __________________________ polygon lies within a circle. 4. Two __________________________ drawn to a circle from an external point are congruent. 5. Name the angle formed by 2 chords which meet on the circle.____________________________ 6. Name a line segment which intersects a circle at 2 points and extends outside the circle._______________ Segment Length Theorems Two Chords:__________________________________ Two Secants:____________________________ _____________________________________________ _______________________________________
Tangent and Secant:________________________________ _________________________________________________ Visualization: **Draw a picture
Chords and intersect at point E within a circle = 9 and = 1
a) if = 6 and = x, write an expression that represents in terms of x
b) find
c) find
AB CD AE EB
CD CE ED
CE
CD
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10R Geometry: Circles HOMEWORK Day 7: Tangents
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10R Circles – Day 8 Segments Practice
Warm-up: Review of Locus x2 + y2 = r2 OR (x - h)2 + (y - k)2 = r2 1. Write an equation of a circle whose center is the origin and radius is 2 2. Write an equation of a circle whose center is (-4, 8) and radius: 5 3. Graph the equation x2 + (y - 1)2 = 9 (x - 2)2 + (y + 1)2 = 1
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Solve for x
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10R Geometry: Circles HOMEWORK Day 8 Practice
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10R Circles – EXTRA PRACTICE
Chords, Tangents, and Secants Practice
3. If AB = 5, BC = 3, and AE = 10 Find AD
Solve for x
Find the value of each variable. (Point O is the center of the circle) 2. 4. 6.
1. If AE EB, CE = 4, and ED = 9
Find AB
5. In the diagram, PQ, QR, RS, and SP are
tangents to O. If PT = 8, QR = 12, and
VS = 5, find the perimeter of PQRS.
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9. Earthquakes: After an earthquake, you are given seismograph readings from three locations, where the coordinate units are miles.
At A(2, 1), the epicenter is 5 miles away.
At B(-2, -2), the epicenter is 6 miles away.
At C(-6, 4), the epicenter is 4 miles away.
a. Graph three circles in one coordinate plane to represent the possible epicenter locations determined by each of the seismograph readings. b. What are the coordinates of the epicenter? c. People could feel the earthquake up to 9 miles from the epicenter. Could a person at (4, -5) feel it? Explain
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Extra Practice
16. / / and m B = 36, find
a) m D
b) m
c) m
d) m A
e) m C
If AB DC
AD
BC
17. Diameter chord at F, is a diameter,
and is a chord of O. If m 60, find:
a) m
b) m A
c) m C
d) m
e) m AOD
f) m
DOE AB AOC
BC BC
AB
AD
CE
18. If is inscribed in a circle so that
m : m : m 2 : 3 : 4
a) mAB
b) mBC
c) mCA
d) m A
e) m B
f) m C
ABC
AB BC CA
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10R Circles – Day 9 Proofs
Circle Theorems All radii All diameters central s arcs arcs central s (Converse) central s chords chords central s (Converse) arcs chords chords arcs (Converse) A diameter to a chord bisects the chord and its arcs
An inscribed is half ½ the arc Inscribed ’s of the same arc are A tangent is to radius at the point of tangency Tangent segments drawn to a from the same external point are Arc midpoints 2 arcs Addition postulate of arcs 2 chords equidistant from the center of a are || lines cut off arcs Inscribed of a semi is a right
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8. 9. 10.
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10R Circles – Day 9 Proofs Homework
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10R Circles – Day 10 Proofs Part II
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1. Given: AC and BC tangent to circle O at A and B. To prove: OC bisects <AOB.
AD BD
2. Given: Secants ADB and CEB intersect at B. AD CE.
To prove: Δ ABC is isosceles.
Statement Reason
Statement Reason
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In the diagram, EF is a tangent segment, measure of arc AD = 140, measure of arc AB = 20, and m<EFD = 60.
a. Find m<CAB b. Prove that ΔABD ~ ΔFEC
Given: PQ and PR tangent to circle O at Q and R. To prove: <PQR <PRQ
Given: Chords AC and BD of circle O intersect at E, AB CD. Prove: ΔABC ΔDCB
Statement Reason
Statement Reason
Statement Reason