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 = __________
(x 2 − 5x)
30
C•
C
•
x
14
40
40
3. x = __________
4. x = __________
(Leave as simplified radical!)
3
x
x
4
C•
8
C•
4
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:
A
5. m CE = _______
13. DQ = ______
6. m∠PCG = _______
14. DG = ______
C
B
F
7. m∠CGP = _______
15. DC = ______
8. CG = ______
16. PG = ______
9. m∠QDC = ________
17. GQ = ______
10. m∠FGD = ________
18. PQ = ______
11. m∠FQD = _______
19. AB = ______
P
E
Q
G
D
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?
X
(c) Find PX.
J
(d) Find PY.
•C
H
Y
P
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.
C•
22.
(a)
(b)
(c)
AB and CD are radii and BD is a common external tangent. AB = 5, CD = 15, BE = 12
Find BD.
B
Find EC.
Find FG.
E
A
A
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.
F
G
B
P
Q
D
C
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
Y
W
X
Tangent relationships are indicated by the diagram. Find the length indicated.
26. JM = 7.1, JK = _______
27. OT=9, OK=15, CD=______
K
C
D
4.5
J
O
T
M
K
28. AB = 10, CD = ________
A
C
B
D
29. AF = FB = 4, DC = 6
find the perimeter of ΔABC
C
D
A
E
F
B
D
C
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 665
1. 120
2. 47
4. 14.04 in
8. No; 52 + 152 162
10. Yes; 62 + 82 = 102
12. 14.2 in
14. 3.6 cm
18. 80.0 km
20. 57.5
22. B
23. All 4 are congruent
25. 35
28. About 5.2 in
Pg 673
2. arcs ET # GH # JN # ML,
segments ET # GH # JN # ML,
angles ‘TFE # ‘HFG # ‘JKN # ‘MKL
4. 2
6. 50
14. 12.5
16. 12—3 | 20.8
17. 108
18. 90
19. 123.855q
26. 8—3 | 13.9 cm
27. 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.)
A
Inscribed Angles Notes
E
™ An angle is inscribed if its vertex is on the circle and
its sides contain chords of the circle.
C
∠_________ is an inscribed angle.
D
B
™ If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its
A
intercepted arc.
∠ABC intercepts _______
q = 100° then m∠ABC = ________.
If m AC
q = ______
If m∠ABC = 70°, then m AC
C
B
™ If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then
the angles are congruent.
∠3 intercepts _______; ∠4 intercepts _______
∠1 intercepts _________
q , ________________
q = m CD
Since m AB
∠2 intercepts _________ so ___________
80
D
B
C
A
1
A
4
3
2
C
B
D
80
A
™ If an inscribed angle of a circle intercepts a
semicircle, then the angle is a right angle.
C
D
B
A
™ The opposite angles of a quadrilateral inscribed in
a circle are supplementary.
N
R
G
™ The measure of an angle formed by a tangent and a chord is
half the measure of the intercepted arc.
m∠C =
1
q
m BDC
2
B
B
●
●
●D
●
●
C
D●
●
●
C
A
1) Name an inscribed angle.
B
2) Name an arc intercepted by ∠BAC
42
P
C
3) If m∠BPC = 42, find m∠BAC
J
4) Find x
L
B
5) RS and TU are diameters of :A .
Find ∠BRT and m∠TRS.
T
R
N
x
100
A
126°
K
S
U
q ≅ RS
q and m∠1 = 38° and m QR
q = 28°. Find:
6) In :A , PQ
m∠T = ________
m∠2 = _________
m∠3 = _________
m∠4 = _________
q = _________
m PT
R
Q
3
4
1
P
S
2
A
T
q = 94, m∠AZB = 104. Find:
7) In :Z , AB & DC , m BC
q = _______
m AB
m∠BAC = _________
m∠ADB = __________
q = ___________
m AD
q = ___________
m CD
m∠DAC = ___________
m∠AEB = ____________
8) Quadrilateral QRST is inscribed in :C . If
m∠T = 95° and m∠S = 100°, find m∠Q and m∠R.
Q
R
●C
A
Z
E
D
C
q =68 and
9) In :Q , AC is a diameter, m CD
q =96. Find:
m BE
m ∠ ABC = _____
m ∠ BDE = _____
m ∠ CED = _____
q = _______
m AD
C
B
Q
A
T
S
B
D
E
Practice: 12-4 Angle Measures only
Name: ___________________________
Date: ________________ Period: _____
Find the measure of each numbered angle.
31.
32.
52°
1
110°
33.
100°
40°
2
3
80°
134°
Given circle T, find the value of x.
34.
35.
36.
130°
20°
100°
T
• T
•
x°
x°
T
70°
•
50°
x°
q =38°. Find:
q = 98°, m OY
q =28°, m q
In :K , m OB
YD =60°, and m DA
q
37. m AB
38. m∠1
B
5
39. m∠2
1
40. m∠3
K
• 2
41. m∠4
42. m∠5
A
4
D
O
3
Y
Name __________________________ Date __________________ Period ____
Chords, Secants and Tangents
Solve for x.
1.
2.
4
x
5
10
3.
4x + 2
8
4
x
x = ______________ 6x - 10
6
x = ______________ 5.
4.
x = ______________ 3x +12
x
6.
x
2
4
5
x = ______________
7.
x = ______________ 8
8.
4
x
x = ______________ 10.
x
4
9
x
3
x
9
6
4
x = ______________
9.
2x – 3
2x – 1
x = ______________
11.
2
x = ______________ 12.
5
10
x
x = ______________ 3
6x - 12
x = ______________ 8
x
3x
6
5
x = ______________
Geometry
Find angles in circles
Name_________________________________
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=__________
H
D
C
O
(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=__________
G
B
E
A
3.
F
A E is a tangent. m∠FAE=85, m∠HJG=55, m∠GAK=75, m AB =40, m BC =16, m CD =10
Find:
G
(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=________
F
L
H
D
J
C
I
B
K
A
E
(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
PreAP/GT Geometry
NOTES
12-5 Equations of Circles
Name ____________________________________
Date _____________Period 1 2 3 4 5 6 7
1. Sketch the graph of the equation.
2
2
( x - 2) + ( y + 4 ) = 36
2. Write the equation of the circle graphed.
________________________________________
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: ax 2 + bx + c = 0
Find the value of c that makes each trinomial a perfect square.
1)
x 2 + 12x + c
2)
x 2 + −7x + c
3)
x 2 + 2x + c
4)
x 2 + 18x + c
5)
t 2 + 40t + c
6)
r 2 − 9r + c
7)
a 2 + 12a + c
8)
h2 − 20h + c
9)
p2 − p + c
5
10) t 2 + t + c
6
Find the exact solution for each equation by completing the square.
11) y2 − 4 y − 5 = 0
12) x 2 + 2x − 143 = 0
13) x 2 − 10x + 21 = 0
14) x 2 + 3x − 18 = 0
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.
2
2
Standard Equation of a Circle
( x - h ) + ( y - k ) = r2
15) x 2 + y2 + 4 x + 6 y − 3 = 0
16) x 2 + y2 + 8x − 2 y + 8 = 0
17) x 2 + y2 − 10x − 12 y − 20 = 0
18) x 2 + y2 − 2x − 4 y + 1 = 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.
x 2 + y 2 − 4 x + 16 y − 13 = 0
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.
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?
a) 100 m
b) 500 m
c) 1 km
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?
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.)
b) If the base of the triangular sleeve is 6.24
inches long, then what is the perimeter of the
triangular sleeve?
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.
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.
b) What is the measure of an inscribed angle in a
garden with a five-pointed star?
15. Graph a circle that contains a diameter with
endpoints (–2, –3) and (4, 5) and then write the
standard equation of the circle.
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.
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 = _____
C
DX = _____
QX = _____
H
TX = _____
CH = _____
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
C
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.
Q
•
B
•P
D
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.
A