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Name: _________________________________________________ Date: _______________________ Period: _________ Homework - Tangents
For each Circle “C”, find the value of “x”. Assume that segments that appear to be tangent are tangent. 1. x = __________ 2. x = __________
• C
2(x 5x)−
14 x
•C
30
40 40
3. x = __________ 4. x = __________ (Leave as simplified radical!) In the figure, AB and CD both are tangent to circle P and circle Q. Also, AP = 8, BQ = 5, and m∠CPE = 45. Find the measure of each of the following: 5. m CE = _______ 13. DQ = ______
6. m∠PCG = _______ 14. DG = ______
7. m∠CGP = _______ 15. DC = ______
8. CG = ______ 16. PG = ______
9. m∠QDC = ________ 17. GQ = ______
10. m∠FGD = ________ 18. PQ = ______
11. m∠FQD = _______ 19. AB = ______
12. m DF = _______
20. PX and PY are tangent to C from an external point P. HJ = 18 and PC = 41. (a) What is the distance from C to X? (b) What is the distance from C to Y? (c) Find PX. (d) Find PY.
21. The minor arc cut off by two tangents to a circle from an outside point is five-sevenths of the major arc. Find the angle formed by the tangents.
4
•C
3 x
x
8•C
4
QPF
E
D
BC
A
G
X
P
J
C
Y H
•C
•
22. AB and CD are radii and BD is a common external tangent. AB = 5, CD = 15, BE = 12 (a) Find BD. (b) Find EC. (c) Find FG.
A C
B
E
D
F G
Find the indicated lengths.
23. Circle P is tangent to each side of ABCD. AB = 20. BC = 11, and DC = 14. Let AQ = x and find AD.
P
A
B
C
D
Q
24. Given: Tangent circles A, B, and C. AB = 8, BC = 13, and AC = 11 Find: The radii of the three circles.
A
C
B
25. Find the perimeter of right triangle WXY if the radius of the circle is 4 and WY = 20
W X
Y
Tangent relationships are indicated by the diagram. Find the length indicated. 26. JM = 7.1, JK = _______ 27. OT=9, OK=15, CD=______
J
K
M
4.5
O
K
T
C D 28. AB = 10, CD = ________ 29. AF = FB = 4, DC = 6 find the perimeter of ΔABC
A B
D
C
FA B
C
D E
Answers to Pg 665 # 1, 2-22 evens (omit # 6 and 16), 23, 25, 28;
Pg 673 # 2, 4, 6, 14, 16-19, 26-27, 30.Pg 6651. 1202. 474. 14.04 in8. No; 52 + 152 � 162
10. Yes; 62 + 82 = 102
12. 14.2 in14. 3.6 cm18. 80.0 km20. 57.522. B23. All 4 are congruent25. 3528. About 5.2 in
Pg 6732. arcs ET � GH � JN � ML,
segments ET � GH � JN � ML,angles �TFE � �HFG � �JKN � �MKL
4. 26. 50
14. 12.516. 12�3 � 20.817. 10818. 9019. 123.855�26. 8�3 � 13.9 cm27. He doesn’t know that the chords are
equidistant from the center.30. 5 in
Review:1. a) Find the measure of the arc CDE.
b) Find the arc length of arc CE.
2. Find the area of the shaded sector in the 2nd circle.
3. Find the area of the shaded segment in the 3rd circle.
4. Find the degree measure of the arc of a sector with area 35� if the area of the circle is 144�.
5. If BC = 2AB, what fraction of the circle is shaded? (Hint: Let the AB = 2x. D is the center of the big circle. AB is the diameter of a little circle and BC is the diameter of a medium circle. Find the areas in terms of x.)
C
A
D
B
EInscribed Angles Notes
An angle is inscribed if its vertex is on the circle and its sides contain chords of the circle.
∠_________ is an inscribed angle.
If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc.
∠ABC intercepts _______
If mAC = 100° then m∠ABC = ________.
If m∠ABC = 70°, then mAC = ______
If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent.
A
C
B
∠3 intercepts _______; ∠4 intercepts _______
Since m∠1 intercepts _________
= mCDAB , ________________ ∠2 intercepts _________ so ___________
If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle.
The opposite angles of a quadrilateral inscribed in a circle are supplementary.
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
m∠C = 1
2m BDC
C
A
D
B
1
2
80B
A4
3 C
D
D
A
C
B
80
N
R
A
G
●
B D
C
●
●
●
D ●
●
●
● C
B
P
C
A B
42
1) Name an inscribed angle.
2) Name an arc intercepted by ∠BAC
3) If m∠BPC = 42, find m∠BAC
4) Find x 5) RS and TU are diameters of .
Find ∠BRT and m∠TRS. A
U
S
A
R
T
B
126°
N
K
J
L
100x
AS
P
T
Q R
12
34
6) In , and m∠1 = 38° and mA PQ RS≅ QR = 28°. Find: m∠T = ________ m∠2 = _________ m∠3 = _________ m∠4 = _________ m = _________ PT 7) In , Z AB DC , mBC = 94, m∠AZB = 104. Find: mAB = _______
Z B
A
D
C
E
m∠BAC = _________ m∠ADB = __________ m = ___________ ADmCD = ___________ m∠DAC = ___________ m∠AEB = ____________ 8) Quadrilateral QRST is inscribed in . If m∠T = 95° and m∠S = 100°, find m∠Q and m∠R.
C
9) In Q , AC is a diameter, mCD =68 and m =96. Find: m
BE∠ABC = _____
m∠BDE = _____ m∠CED = _____ m = _______ AD
A
Q
CB
ED
Q R
T S
● C
Practice: 12-4 Angle Measures only Name: ___________________________ Date: ________________ Period: _____ Find the measure of each numbered angle.
31. 32. 33. Given circle T, find the value of x.
34. 35. 36. In , mK OB = 98°, m =28°, mOY YD =60°, and mDA =38°. Find: 37. m AB 38. m∠1 39. m∠2 40. m∠3 41. m∠4 42. m∠5
40° 52° 1
134°
2
100°
110°
3
80°
•
130°
x° •T
50° •T
20°
x° 70°
T 100°
x°
B
O
Y • K
A D
1
2
3
4
5
Name __________________________ Date __________________ Period ____
Chords, Secants and Tangents Solve for x.
1.
2. 3.
4.
5. 6.
7.
8. 9.
10.
11. 12.
4x + 2 x 4 8 10 4 5
x 6 6x - 10
x = ______________ x = ______________ x = ______________
3x +12 x x 2
4 3 4 5 6x - 12
x = ______________ x = ______________ x = ______________
2x 8 3 x – 3
2 2x – 1
6 9 4
x
x = ______________ x = ______________ x = ______________
10 5
5 x
4 x x 8
3x 6 x 9
x = ______________ x = ______________ x = ______________
Geometry Name_________________________________ Find angles in circles Date_____________________Period_______ 2. DB is a diameter of circle O. ED and EA are tangents of circle O
m AB =76 and m DC =110. Find:
(a) m BC =__________ (i) m∠EAD=__________
(b) m AD =__________ (j) m∠BAF=__________
(c) m∠DOA=__________ (k) m∠DCA=__________
(d) m∠DAO=__________ (l) m∠DGC=__________
(e) m∠OAC=__________ (m) m∠DBA=__________
(f) m∠CAB=__________ (n) m∠ADO=__________
(g) m∠EDA=__________ (o) m∠ODC=__________
(h) m∠DEA=__________ (p) m∠HDC=__________
3. A E is a tangent. m∠FAE=85, m∠HJG=55, m∠GAK=75, m AB =40, m BC =16, m CD =10
Find:
(a) m DF =_________ (i) m∠HAG=________
(b) m FG = _________ (j) m∠GAF=________
(c) m GH =_________ (k) m∠ALC=________
(d) m HA =_________ (l) m∠FIB=_________
(e) m∠AFD=________ (m) m∠KEH=________
(f) m∠AGC=________ (n) m∠HEG=________
(g) m∠AHB=________ (o) m∠GEF=________
(h) m∠KAH=________
Answers:
2: (a) 70 (b) 104 (c) 104 (d) 38 (e) 17 (f) 35 (g) 52 (h) 76 (i) 52 (j) 38 (k) 52 (l) 93 (m) 52 (n) 38 (o) 35 (p) 55
3: (a) 104 (b) 40 (c) 70 (d) 80 (e) 33 (f) 28 (g) 20 (h) 40 (i) 35 (j) 20 (k) 48 (l) 105 (m) 20 (n) 27 (o) 15
B
D
H
A
C
O
G
F
E
G F
D
E
H
A
L
C
B I
J
K
PreAP/GT Geometry NOTES Name ____________________________________ 12-5 Equations of Circles Date _____________Period 1 2 3 4 5 6 7 1. Sketch the graph of the equation. 2. Write the equation of the circle graphed. ( ) ( )2 2x - 2 + y + 4 = 36 ________________________________________
3. Write the equation of the line tangent to the circle in question #2 at the point: a. (-4, 2)
b. (1, -3)
c. (0, 0)
d. (-1, -7)
e. (-7, 1)
f. (-8, 0)
4. Finding the equation of the circle given 3 points on the circle. Write the equation of the circle that contains the points X( - 6, - 1), Y( - 4, 3), and Z(2, - 5). A) Write the equations for the perpendicular bisectors for Δ XYZ. Tell what segments you used. B) Find the intersection of the two perpendicular bisectors. This will be the circumcenter (the center of the circumscribed circle). (Hint: solve as a system of equations.) C) Determine the radius of the circle. Write the formula and substitution step. D) Write the equation of the circle through X, Y, and Z.
Name _____________________________________ Date _______________________ Period ______ Worksheet - Completing the Square PreAP Geometry
Standard Form of a quadratic equation: 2 0ax bx c+ + = Find the value of c that makes each trinomial a perfect square.
1) 2) 2 12x x+ + c c2 7x x+ − + 3) 4) 2 2x x+ + c c2 18x x+ + 5) 6) 2 40t t+ + c c2 9r r− + 7) 8) 2 12a a+ + c c2 20h h− +
9) 10) 2p p− + c 2 56
t t c+ +
Find the exact solution for each equation by completing the square.
11) 12) 2 4 5y y− − = 0 2 2 143 0x x+ − = 13) 14) 2 10 21 0x x− + = 2 3 18 0x x+ − = Let’s try completing the square for equations of circles. Rewrite the equation in standard form. Solve for the center of the circle and the radius. Standard Equation of a Circle ( ) ( )2 2 2x - h + y - k = r 15) 16) 2 2 4 6 3x y x y+ + + − = 0 02 2 8 2 8x y x y+ + − + = 17) 18) 2 2 10 12 20 0x y x y+ − − − = 2 2 2 4 1x y x y 0+ − − + =
Circles Review and Word Problems Name: ___________________________ Date: ________________ Period: _____ Show all work! This is due the day of the test. Give exact answers if possible. If not, round to nearest tenth. 1. Carl is planning to visit a circular park. The radius
of the park is 8 miles. He is looking at a map of the park and sees that the park has five landmarks along its edge. The landmarks are connected by paths of equal length for biking. These paths form a regular pentagon inscribed in the circle. If Carl bikes along these paths to visit each landmark, how many miles will he bike?
2. A circle is inscribed in a 40°-60°-80° triangle. The points of tangency form the vertices of a triangle inscribed in the circle. What are the angles of the inscribed triangle?
3. Cora is wrapping a ribbon around a cylinder-shaped gift box. The box has a diameter of 15 inches and the ribbon is 60 inches long. Cora is able to wrap the ribbon all the way around the box once, and then continue so that the second end of the ribbon passes the first end. What is the central angle formed between the ends of the ribbon? Round your answer to the nearest tenth of a degree.
4. A wheel is rolling down an incline. Twelve evenly spaced diameters form spokes of the wheel. When spoke 2 is vertical, which spoke will be perpendicular to the incline?
5. Vanessa looked through her telescope at a mountainous landscape. The figure shows what she saw. Based on the view, approximately what angle does the side of the mountain that runs from A to B make with the horizontal?
6. Complete the square to write the equation of the circle in standard form. Then give the location of the center and the radius.
013 16422 =−+−+ yxyx
7. Francisco is a painter. He places a circular canvas on his A-frame easel and carefully centers it. The apex of the easel is 30° and the measure of arc BC is 22°. What is the measure of arc AB ?
8. A geostationary satellite is about 35,800 kilometers above Earth. How many arc degrees of the planet are visible to a camera in the satellite?
9. The circle below represents Earth. The radius of Earth is about 6400 km. Find the distance d that a person can see on a clear day from each of the following heights h above Earth. Round your answer to the nearest tenth of a kilometer. a) 100 m b) 500 m c) 1 km
10. Archeologists and scientists unearthed part of a circular wall. They made the measurements shown in the figure. Based on the information in the figure, what was the radius of the circle?
11. The figure shows the cross-section of an axle held in place by a triangular sleeve. A brake extends from the apex of the triangle. When the brake is extended 2.5 inches into the sleeve, it comes into contact with the axle. a) What is the diameter of the axle? b) If the base of the triangular sleeve is 6.24 inches long, then what is the perimeter of the triangular sleeve?
12. The diameter of the base of a cylindrical milk tank is 59 in. The length of the tank is 470 in. You estimate that the depth of the milk in the tank is 20 in. Find the number of gallons of milk in the tank to the nearest gallon. (1 gal = 231 in.3) (Diagram is not to scale.)
Hint: First find the length of the chord, then the area of the sector, and subtract the area of the triangle.
13. Some circular English gardens, like the one shown here, have paths in the shape of an inscribed regular star. a) Find the measure of an inscribed angle formed by the star in the garden shown here.
b) What is the measure of an inscribed angle in a garden with a five-pointed star?
14. The radius of Earth’s equator is about 3960 miles. a) Write the equation of the equator with the center of Earth as the origin. b) Find the length of a 1° arc on the equator to the nearest tenth of a mile. c) A 1° arc along the equator is 60 nautical miles long. How many miles are in a nautical mile? Round to the nearest tenth. d) Columbus planned his trip to the East by going west. He thought each 1° arc was 45 miles long. He estimated that the trip would take 21 days. Use your answer to part (b) to find a better estimate.
15. Graph a circle that contains a diameter with endpoints (–2, –3) and (4, 5) and then write the standard equation of the circle.
16. In the circle with center D, EX = 24, DE = 7, XT is a secant, EX and AX are tangents, and CH is the perpendicular bisector of AD . Find each measure. AX = _____ DX = _____ QX = _____ TX = _____ CH = _____
H
C
Find the value of x to the nearest hundredth. Assume that segments that appear tangent are tangent. 17.
18.
19.
20.
21. Find each angle measure. SHOW ALL WORK! (for example, write 100+20, 70/2, 180-113, etc.)
T
22. Find each angle measure. SHOW ALL WORK!
D
23. In right triangle ABC, CD is an altitude. The circles centered at P and Q are inscribed in triangles ACD and BCD, respectively. For AC = 15 and BC = 20, compute PQ.
A B
C
D
• • P
Q Answers: 1. 47.02 miles. 2. 50°, 60°, 70° 3. 98.366°. 4. spoke 10. 5. 60°. 6. (x-2)2 + (y+8) 2 = 81; center (2, -8); radius 9. 7. 128°. 8. 162.6°. 9. a) 35.8 km, b) 80.0 km, c) 113.1 km. 10. 25.5 ft. 11. a) 3.9 in. b) 20.48 in. 12. 1661 gal. 13. a) 77.1, b) 36. 14. a) x2 + y2 = 15,681,600. b) 69.1 mi. c) 1.2 mi. d) about 32 days. 15. (x – 1)2 + (y – 1)2 = 25. 16. AX = 24, DX = 25, QX = 18, TX = 32, CH = 7√3 ≈ 12.12. 17. 28.05. 18. 3. 19. 4.67. 20. 5.66. 21. m∠1 = 70, m∠2 = 35, m∠3 = 55, m∠4 = 30, m∠5 = 55, m∠6 = 50, m∠7 = 75, m∠8 = 25, , m∠9 = 80, m∠10 = 45. 22. m∠1 = 52, m∠2 = 22, m∠3 = 74, m∠4 = 40, m∠5 = 44, m∠6 = 40, m∠7 = 66, m∠8 = 56, m∠9 = 65, m∠10 = 49, m∠11 = 35, m∠12 = 65, m∠13 = 31, m∠14 = 44, m∠15 = 84. 23. 5√2.
