Chapter 9 Circles
Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates, theorems and corollaries in this chapter. D. Recognize circumscribed and inscribed polygons. E. Prove statements involving circumscribed and inscribed polygons. F. Solve problems involving circumscribed and inscribed polygons. G. Understand and apply theorems related to tangents, radii, arcs, chords, and central angles.
Section 9-1 Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18
Objectives A. Understand and apply the terms circle, center, radius, chord, secant, diameter, tangent, point of tangency, and sphere. B. Understand and apply the terms congruent circles, congruent spheres, concentric circles, and concentric spheres. C. Understand and apply the terms inscribed in a circle and circumscribed about a polygon. D. Correctly draw inscribed and circumscribed figures.
Circular Logic • On a piece of paper, accurately draw a circle. • What method did you use to make sure you drew a circle? Circle set of all coplanar points that are a given distance (radius) from a given point (center). – Basic Parts: • Radius Distance from the center of a circle to any single point on the circle. • Center Point that is equidistant from all points on the circle. • Indicated by symbol – P Circle with center P • Contrast a circle to a sphere: – Sphere set of all points in space a given distance (radius) from a given point (center)
Circle
Center
Radius
Lines and Line Segments Related to Circles Chord segment whose endpoints lie on a circle. Diameter a chord that passes through the center. Secant line that contains a chord. Tangent line in the plane of a circle that intersects the circle at exactly one point. Point of Tangency the point of intersection between a circle and a tangent to the circle.
Segments & Lines Chord
Diameter
Secant
Tangent Point of Tangency
Circular Relationships • • • •
Concentric circles coplanar circles with the same center Concentric Spheres spheres with the same center Congruent Circles circles with congruent radii Congruent Spheres spheres with congruent radii
Concentric Circles
Congruent Circles
A Figure Within a Figure Circumscribed About A Polygon Inscribed In a Circle all of the vertices of the polygon lie on a circle
Circumscribed About A Polygon
Inscribed In A Circle
Sample Problems 1. Draw a circle and several parallel chords. What do you think is true of the midpoints of all such chords?
Sample Problems 3. Draw a right triangle inscribed in a circle. What do you know about the midpoint of the hypotenuse? Where is the center of the circle? If the legs of the right triangle are 6 and 8, find the radius of the circle.
8
6
Sample Problems 5. The radii of two concentric circles are 15 and 7. A diameter AB of the larger circle intersects the smaller circle at C and D. Find two possible values for AC.
Sample Problems 7. Draw a circle with an inscribed trapezoid.
Sample Problems Draw a circle and inscribe the polygon named. 9. a parallelogram 11. a quadrilateral PQRS with PR a diameter
Sample Problems For each draw a O with radius 12. Then draw OA and OB to form an angle with the measurement given. Find AB. 13. m AOB = 180 15. m AOB = 120 17. Q and R are congruent circles that intersect at C and D. CD is the common chord of the circles. What kind of quadrilateral is QDRC? Why? CD must be the perpendicular bisector of QR. Why? If QC = 17 and QR = 30, find CD. C
Q
R
D
Section 9-2 Tangents Homework Pages 335-337: 1-18 Excluding 14
Objectives A. Understand and apply the terms “external common tangent” and “internal common tangent”. B. Understand and apply the terms “externally tangent circles” and “internally tangent circles”.
C. Understand and apply theorems and corollaries dealing with the tangents of circles.
External Common Tangent • External Common Tangent a line that is tangent to two coplanar circles and doesn’t intersect the segment joining the centers of the circles.
Internal Common Tangent • Internal Common Tangent a line that is tangent to two coplanar circles and intersects the segment joining the centers of the circles .
Tangent Circles • Externally Tangent Circles coplanar circles that are tangent to the same line at the same point and the centers are on opposite sides of the line. • Internally Tangent Circles coplanar circles that are tangent to the same line at the same point and the centers are on the same side of the line.
Externally Tangent Circles
Externally Tangent Circles
Center Center
Internally Tangent Circles
Internally Tangent
Center Center
Point of Tangency
Theorem 9-1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency
Theorem 9-1 Corollary 1 Tangents to a circle from a point are congruent.
Theorem 9-2
tangent line
If a line in a plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle.
Sample Problems JT is tangent to O at T. 1. If OT = 6 and JO = 10, JT = ? 3. If m TOJ = 60 and OT = 6, JO = ?
O K J
T
5. The diagram shows tangent lines and circles. Find PD.
A
8.2 P D B
C
Sample Problems 7. What do you think is true of the common external tangents AB and CD? Prove it. Will the results to this question still be true if the circles are congruent?
B A
Z C D
Sample Problems 9. Draw O with perpendicular radii OX and OY. Draw tangents to the circle at X and Y. If the tangents meet at Z, what kind of figure is OXZY? Explain. If OX = 5, find OZ. 11. Given: RS is a common internal tangent to A and B. Explain why AC RC BC SC R A
B
C S
Sample Problems 13. State the theorem which would describe the relationship between the planes tangent to a sphere at either end of a diameter. 15. PA, PB, and RS are tangents. Explain why PR + RS + SP = PA + PB A
R C
B
P S
Sample Problems 17. JK is tangent to P and Q. JK = ? J K
P
11
3
3 Q
Sample Problems 19. Given two tangent circles; EF is a common external tangent. Prove something about G. Prove something about EHF.
E G F H
Section 9-3 Arcs and Central Angles Homework Pages 341-342: 1-20 Excluding 12
Objectives A. Understand and apply the term “central angle”. B. Understand and apply the terms “major arc”, “minor arc”, “adjacent arcs”, “congruent arcs”, and “intercept arc”.
C. Understand and utilize the Arc Addition Postulate. D. Understand and apply the theorem of congruent minor arcs.
Central Angle an angle whose vertex lies on the center of a circle. Central Angle
Not Noah’s ‘Arc’ Arc an unbroken part of a circle. • Types of arcs: • Major arc • Minor arc • Adjacent arcs • Congruent arcs • Intercepted arc (covered in section 9-5) • The symbol for the measurement of an arc is:
mAB measurement of arc AB
Adjacent Arcs arcs of the same circle that have exactly one point in common .
Congruent Arcs arcs in the same circle or congruent circles that have the same measurement.
Congruent Arcs
Same Length
Intercepted Arc the arc between the sides of an inscribed angle
Intercept Arc
Inscribed Angle
Minor Arc an unbroken part of a circle that measures less the 180°.
Measure of a minor arc = measure of its central angle.
79
Measure of a minor arc = 79
Major Arc an unbroken part of a circle that measures more than 180° and less than 360°.
Measure of major arc = 360 - measure of the minor arc
360 109 251
109
Semicircle an unbroken part of a circle that measures exactly 180 degrees.
Semicircle arcs whose endpoints are the endpoints of a diameter.
Postulate 16 – Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs. B
A
mAB m BC mAC
C
Theorem 9-3 In the same circle or congruent circles, two minor arcs are congruent if and only if their central angles are congruent.
Sample Problems Find the measure of the central 1. 1.
85°
3.
1
1
5.
240°
68° 1
150°
Sample Problems 7. At 11 o’clock the hands of a clock form an angle of ? 9. Draw a circle. Place points A, B and C on it in such positions that mA B mB C mA C
Sample Problems
OC, OB, and OA are all radii. So OC = OB = OA
C
If mCOB 42, then mCOA 138.
A
O
B
Since OC OA, then AOC is isosceles.
Since AOC is isosceles, mACO mCAO.
D
mAOC mCAO mACO 180 138 (2 mCAO ) 180 (2 mCAO ) 42 mCAO 21
Sample Problems
mC B
mB D
70
60
66
60
p
30
28
?
?
q
mCOD
?
?
100
?
?
mCAD
?
?
?
52
?
C
A
O
B D
Sample Problems W
15. Given: WZ is a diameter of O;
mW X mX Y n Prove: m Z = n
O
Z
X Y
The latitude of a city is given. Find the radius of this circle of latitude. 17. Milwaukee, Wisconsin; 43°N 19. Sydney, Australia; 34°S
Section 9-4 Arcs and Chords Homework Pages 347-348: 1-22
Objectives A. Understand the term ‘arc of a chord’. B. Understand and apply theorems relating arcs and chords to circles. C. Use the theorems related to arcs and chords to solve problems involving circles.
Arc of a Chord • Arc of a Chord the minor arc created by the endpoints of the chord. Arc of a Chord Chord
Theorem 9-4 In the same circle or congruent circles: (1) Congruent arcs have congruent chords. (2) Congruent chords have congruent arcs.
Theorem 9-5 A diameter that is perpendicular to a chord bisects the chord and its arc.
Theorem 9-6 In the same circle or congruent circles: (1) Chords equally distant from the center (or centers) are congruent. (2) Congruent chords are equally distant from the center (or centers).
A
E
B
F
C
D
AB = CD = EF
Sample Problems Y
M X
3 5 O
OMY is a pattern right triangle. Therefore, MY 4. A diameter that is perpendicular to a chord bisects the chord and the arc.
1. XY = ? XY = 2MY = 2(4) = 8
Sample Problems
O T S 3. OT = 9, RS = 18 OR = ?
R
Sample Problems A B
A B
O O
C
5. mB C
C
D
D ?
360 120 80 3
Sample Problems B
A
O 7. m AOB = 60; AB = 24 OA = ?
M
A O D
B C
N 9. AB = 18; OM = 12 ON = 10; CD = ?
Sample Problems 11. Sketch a circle O with radius 10 and chord XY, 8. How far is the chord from O? 13. Sketch a circle P with radius 5 and chord AB that is 2 cm from P. Find the length of AB. 15. Given: J K Prove: J Z K Z Z
J
K
Sample Problems K
17. OJ = 10, JK = ? 120°
O J
19. A plane 5 cm from the center of a sphere intersects the sphere in a circle with diameter 24 cm. Find the diameter of the sphere. 21. Use trigonometry to find the measure of the arc cut off by a chord 12 cm long in a circle of radius 10 cm.
Section 9-5 Inscribed Angles Homework Pages 354-356: 1-24 (no 14)
Objectives A. Understand and apply the terms “inscribed angle” and “intercepted arc”. B. Understand and apply the theorems and corollaries associated with inscribed angles and intercepted arcs of circles. C. Use the theorems and corollaries associated with inscribed angles and intercepted arcs to solve problems involving circles.
Inscribed Angle Inscribed Angle an angle whose vertex lies on a circle and whose sides contain chords of the circle.
Chords
Vertex
Inscribed Angle
Intercepted Arc Intercepted Arc an arc formed on the interior of an angle.
Intercepted Arc
Inscribed Angle
Theorem 9-7 The measure of an inscribed angle is equal to half the measure of its intercepted arc.
½(x ) x
Theorem 9-7 Corollary 1 If two inscribed angles intercept the same arc, then the angles are congruent.
Theorem 9-7 Corollary 2 An angle inscribed in a semicircle is a right angle.
Theorem 9-7 Corollary 3 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. B m A + m C = 180 m B + m D = 180
A
C D
Theorem 9-8 The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.
1 m ABC = m A B 2 A
B C
Sample Problems 1.
100°
x°
z° y°
O
3. x°
50°
y° z°
80°
120°
70°
What else do you know? Is there a diameter? What is the measure of a semicircle? 100° + x° + 50° = 180° Why? x° = 30° What else do you know? The measure of an inscribed angle is equal to half the measure of its intercepted arc. y° = ( ½ ) 50° y° = 25° z° = ( ½ ) 30° z° = 15° What else do you know? If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. 70° + x° = 180° x° = 110° 80° + y° = 180° y° = 100° What else do you know? 120° + z° = 2(x°) Why? 120° + z° = 2(110°) z° = 100°
Sample Problems
5. y°
50°
z°
z°
x° 100°
What else do you know? Why? z° + z° + 50° = 180° Why? z° = 65° Why? What else do you know? Why? The measure of an inscribed angle is equal to half the measure of its intercepted arc. What else do you know? The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. x° = ( ½ ) 100° x° = 50° y° = ? z° = ( ½ ) y° 65° = ( ½ ) y° y° = 130°
Sample Problems
7. x°
What else do you know? If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. y° x° + 76° = 180° x° = 104° What else do you know? 76° In the same or congruent circles, y° congruent chords have congruent arcs. What else do you know? z° 2x° = y° + y° Why? y° = 104° Why? What else do you know? The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.
z° = ( ½ ) y° z° = ( ½ ) 104° z° = 52°
Sample Problems
9.
z° x°
y° x°
90°
What else do you know? Why? What else do you know? The measure of an inscribed angle is equal to half the measure of its intercepted arc. x° = ( ½ ) 100° x° = 50° y° = ? ( ½ ) y° = x° ( ½ ) y° = 50° y° = 100° z° = ? z° = ( ½ ) (360° - (100°+ 90° + 100°)) Why? z° = 35°
Sample Problems 17. Draw an inscribed quadrilateral ABCD and its diagonals intersecting at E. Name two pairs of similar triangles.
Sample Problems ABCD is an inscribed quadrilateral. 19. m A = x, m B = 2x, m C = x + 20. Find x and m D. 21. m D = 75, mA B x 2 , mB C 5x and mC D 6x. Find x and m A. 23. Equilateral ABC is inscribed in a circle. P and Q are midpoints of arcs BC and CA respectively. What kind of figure is quadrilateral AQPB? Why?
Section 9-6 Other Angles Homework Pages 359-360: 1-24
Objectives A. Understand and apply the theorem relating to two chords intersecting inside of a circle. B. Understand and apply the theorem relating two secants, two tangents or a secant and a tangent of a circle. C. Use these theorems to solve problems relating to circles.
Theorem 9-9 The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. C A
x
x D
B
mAB m C D 2
Theorem 9-10 The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs. A
A
B A
B C x
x
D
C
x
mAB m C D 2
x
B
mAB m B C 2
x
x
C
mA B C m A C 2
Hints to help you remember these theorems! • If the vertex of the angle in question is INSIDE of the circle, ADD the intercepted arcs and divide by two to get the measure of the angle. • Can a measure of an angle EVER be negative? • If the vertex of the angle in question is OUTSIDE of the circle, SUBTRACT the smaller intercepted arc from the larger intercepted arc and divide the result by two to get the measure of the angle.
ADD! R
SUBTRACT! U
A
B
1 T
S
mRT mUS m1 2
C mCT mBT mA 2
T
Sample Problems 1- 10: Find the measure of each angle. What should you do first? BZ is a tangent line. AC is a diameter. m BC 90 m CD 30 m DE 20
A
B
9 m9 ? m AB ?
m1 ? 1 3 m3 ? O
Z 10 90°
8 4
m1 90 m3 25 m5 55 m9 90 m7 35 C
What type of angle 30° m5 ? 5 is angle 1? 6 D What type of angle is angle 3? 1 m 5 20 90 1 2 20° 1 72 m3 30 20 m 90 20 2 2 E Where is the Where is the 7 m7 ? vertex of angle 5? vertex of angle 7?
Sample Problems Complete. 11. If m RT 80 and mUS 40, then m 1 = ? 13. If m 1 50 and m RT 70, then mUS ?
R
U 50° 1
80° 70°
T
40°
S
# 11 What should you do first? Where is the vertex of the angle? The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.
# 13 What should you do first? Where is the vertex of the angle? The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.
m RT mUS 80 40 m1 60 2 2
m1
m RT mUS 2
100 70 mUS
70 mUS 2 mUS 30
50
Sample Problems
A
Segment AT is a tangent line. 15. If mCT 110 and m BT 50, then m A = ?
B 50°
C
T
110° # 15 What should you do first? mCT m BT 110 50 What else do you know? m A 30 2 2 What type of lines contain segments AC and AT? Where is vertex of the angle? The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.
Sample Problems
A
Segment AT is a tangent line. 17. If m A 35 and mCT 110,
B
35°
then mBT ?
T
C # 17 What should you do first? What else do you know? What type of lines contain segments AC and AT? Where is the vertex of the angle? The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.
110° mCT m BT 2 110 m BT 35 2 m A
70 110 m BT m BT 40
Sample Problems PX and PY are tangent segments. X 19. If mXY 90, then mP ?
P
Z
90°
Y
The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal mXZY ? to half the difference of the measures of mXZY 360 mXY 360 90 270 the intercepted arcs. Where is the vertex of the angle? m XZY m XY 270 90 m P 90 What else do you know? 2 2 # 19 What should you do first? What else do you know?
Sample Problems PX and PY are tangent segments. X 21. If mP 65, then mXY ?
P
Z
65°
Y
The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.
# 21 What should you do first? What else do you know? mXZY ? mXZY 360 mXY
Where is the vertex of the angle? What else do you know? 360 mXY m XY 65 2
m P
m XZY m XY 2
130 360 2mXY
2mXY 230
mXY 115
Sample Problems 23. A quadrilateral circumscribed about a circle has angles 80, 90, 94 and 96. Find the measures of the four nonoverlapping arcs determined by the points of tangency. 27. Write an equation c° involving a, b and c. b° a°
Section 9-7 Circles and Lengths of Segments Homework Pages 364-366: 1-26
Objectives A. Understand and apply theorems relating the product of segments of chords, secants, and tangents of a circle. B. Use these theorems to solve problems involving circles.
Theorem 9-11 When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.
x a
(a) (b) = (x) (y) b
y
Theorem 9-12 When two secants are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. a b y x
(a) (b) = (x) (y)
Theorem 9-13 When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment equals the square of the tangent segment a
b (a) (b) = x2 x
Sample Problems – Solve for x. 1.
5 4
x 8
3. 4
# 1 What else do you know? When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. 4(x) = 5(8) x = 10
# 3 What else do you know? When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment equals the square of the tangent segment.
x2 7 3 x 21
3 x
Sample Problems – Solve for x. # 5 What else do you know?
5
3
5.
4 x
When two secants are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.
(8)(5) = (4 + x)(4) 40 = 16 + 4x
7. x 6
5 4
24 = 4x x = 6
# 7 What else do you know? When two secants are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. (x)(5) = (10)(4)
x=8
Sample Problems – Solve for x. 9. 10 3x
x
# 9 What else do you know? When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment equals the square of the tangent segment.
10 3x x x 2
100 4x x
100 4x 2 25 x 2 x 25
x 5
x5
Sample Problems Chords AB and CD intersect at P. 13. AP = 6, BP = 8, CD = 16 DP = ?
A
D 6
P
8 16
# 13 What should you do first? What else do you know? B C When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. (6)(8) = (DP)(16 – DP) 48 16DP DP2 Now what? DP2 16DP 48 0
b b 2 4ac (16) (16) 2 4(1)(48) 16 256 192 DP DP DP 2a 2(1) 2 16 8 16 64 24 8 DP DP 12 or 4 DP DP or 2 2 2 2
Sample Problems A
Chords AB and CD intersect at P. 15. AB = 12, CP = 9, DP = 4 BP = ? # 15 What should you do first? What else do you know? When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. (4)(9) = (BP)(12 – BP) 36 12BP BP2
b b 2 4ac BP 2a
BP
12 144 144 2
9
D P
4
12
C
B
BP2 12BP 36 0
(12) (12) 2 4(1)(36) BP 2(1) 12 BP 6 2
Sample Problems T
Segment PT is tangent to the circle. 17. PT = 6, PB = 3 AB = ? A
B D
# 17 What should you do first? What else do you know?
When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment equals the square of the tangent segment.
6
2
3 AP
AP AB BP
36 3AP 12 AB 3
6 3
C
AP 12 AB 9
P
Sample Problems T
Segment PT is tangent to the circle. 19. PD = 5, CD = 7, AB = 11, PB = ? A
P
11
5
B D
# 19 What should you do first? What else do you know?
7
C When two secants are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.
BP AP DP CP BP 11 BP 512 BP2 11BP 60 0 BP 15 BP 4 0 Can BP = -15? Can BP = 4?
BP = 4
11BP BP2 60 BP 15 or 4
Sample Problems 23. A bridge over a river has the shape of a circular arc. The span of the bridge is 24 meters. The midpoint of the arc is 4 meters higher than the endpoints. What is the radius of the circle that contains the arc.
Chapter 9 Circles Review Homework Page 371: 2-16 evens
