Download 6-1 Basic Definitions and Relationships (Day 1)

6-1 Basic Definitions and Relationships (Day 1)
Define each of the following terms and draw an example for each.
If you do not already know a definition, read section 6-1 to help you.
Term
Circle
Definition
Diagram
Radius
Diameter
Chord
Secant
Tangent
Concentric circles
Congruent circles
Central Angle
Inscribed Angle
Inscribed polygon
Circumscribed polygon
1
Minor arc
Semicircle
Major arc
Intercepted arc
Classwork Problems:
Do p. 225 (top of page) Oral Exercises #3-12.
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
Do p. 225 Written Exercises #1-14
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
End
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6-2 Inscribed Angles (Day 2)
1. Consider O below.
 (Hint: Draw in line segment ____)
a. If m∠BAC = 50°, find m BC

b. If m∠BAC = 20°, find m BC
.
c. If m∠BAC = x°, find m BC
B
A
C
O
Inscribed Angle Theorem
The measure of an inscribed angle is equal to
2. Consider O below.
?
a. Suppose that m∠1 = 30°. What is m BD
.
b. Suppose that m∠1 = 30° and m∠2 = 20°. Find m∠BAC and m BC
.
c. Suppose m∠1 = 15° and m∠2 = 75°. Find m∠BAC and m BC
?
d. If m∠BAC = 120°, what is m BC
B
?
e. If m∠BAC = x°, what is m BC
A
1
2
O
D
C
3
 , mAB
 , and mBDC
.
3. In the diagram, AB ≅ AC . Find mAC
A
B
20°
C
D
 = 200° . Find x and y.
4. In the diagram mACB
B
x
y
A
20°
C
5. Consider the circle to the right.
a. Find x.
Corollary – Inscribed angles that intercept the
same arc or congruent arcs
B
x
70°
C
P
.
b. Find m BC
100°
A
D
c. Explain why ΔABP ~ ΔDCP.
d. Explain why P could not be the center of this circle.
 = 140° and mBCD
 = 180°.
6. In the circle below, m AC
A
a. Find x.
b. Find y, z, and m∠BEA.
30°
B
E
z
x
D
y
C
4
7.
AB is a diameter of a circle with center O. Let P be any other point on the circle. Find m∠APB.
Corollary – An angle inscribed in a semicircle
8. To the right a quadrilateral is inscribed inside a circle.
a. Find x.
A
z
x
b. What can be said about the opposite angles of a
quadrilateral that is inscribed in a circle?
B
y
D
c. If z is double y, find y and z.
C
40°
Corollary – Opposite angles of a quadrilateral inscribed in a circle are
d. How would you algebraically prove the corollary above?
 = 40° . You may assume that AB ⊥ BD
9. In the diagram below, BA is tangent to O at B, and mBC
. (You will prove this algebraically in a following problem.)
.
a. Find mCD
D
b. Find m∠DBC.
c. Find m∠CBA.
O
C
B
A
5
D
 =78°.
10. Repeat the previous problem with mBC
O
C
 =2x, then m∠CBA = x.
11. Show that if mBC
B
A
Theorem: If a tangent and a radius (or diameter) intersect at the point of tangency, then
and
12. Draw a parallelogram inscribed in a circle. What is the only kind of parallelogram that you can draw?
13. Recall:
a. What is always true about the opposite angles of a parallelogram?
b. What can be said about the opposite angles of a quadrilateral inscribed in a circle?
c. Explain mathematically why a parallelogram inscribed in a circle must be a rectangle.
14. A parallelogram with perimeter 164 is inscribed in a circle of radius 29. Find the area of the
parallelogram.
End
6
6-3 Tangents (Day 3)
1. In the diagram below, AB is tangent to the circle at C.
 =210° and mCED
 =260°,
If mCDE
 . (Note: It’s not 50°.)
a. Find mDE
b. Find x, y, and z.
2. Line l is tangent to the circle below at C. If m∠ACD = m∠D
 = 1 mADC
 , find x and y.
and mAB
5
Corollary: The measure of an angle formed by a tangent and a chord is equal to
3. In the circle below, AC and BC are tangents. In this
problem, you will show that AC = BC.
a. Lightly draw in AB.
 =100°. Find m∠BAC and
b. Suppose that mAB
m∠ABC.
 =120°. Find m∠BAC and
c. Suppose that mAB
m∠ABC.
d. Why must ∠BAC and ∠ABC always be equal?
e. Why must AC = BC?
7
Theorem: The segments of two tangents to a circle from a common external point
 =200°, find m∠C.
4. Consider again the diagram from #3. If mADB
5. Below is a circle inscribed in ΔABC. AB = 20 and BC = 17. Find the lengths of EB, BF, FC, CD,
AD, and AC.
6. A circle is inscribed in a triangle in the diagram below. If AC = 21, find x.
7. A triangle is circumscribed about a circle below. A, B, and C are points of tangency. If AD = 10x +
y, CD = 4x – y, BF = 4x, and CF = 2 – y, find x and y.
8
8. Below O is inscribed in ΔABC. D, E, and F are points of tangency. ΔABC is isosceles with AB =
AC = 10 and BC = 12.
a. Find BD and DA.
b. Find AE.
c. Why is ΔAOD ~ ΔABE?
d. Find the radius of the circle, r.
e. Find the length of chord DF.
f. Find the distance from DF to the center of the
circle.
Selected answers: d. 3, e. 24/5, f. 9/5
9. In the diagram below, AB is tangent to both circles. RS = 8, P has radius 8, and Q has radius 5.
a. Why is PA  QB ?
b. Find AB. (Hint: Draw a segment from Q to AP that is parallel to AB .)
10. A circle is inscribed in a 5-12-13 right triangle. Find its radius. (DYOD)
End
9
6-4 Angles Inside and Outside Circles (Day 4)
Problems 1 and 2 consider the relationship between the angle formed by intersecting chords and the
measure of their intercepted arcs.
1. Consider the diagram below. Draw auxiliary line BD .
a. Find m∠DBE.
b. Find m∠BDE.
c. Find x.
2. Repeat problem 1 with
 = 80° and mCD
 = 90°
a. mAB
 = w and mCD
 = z.
b. mAB
 and CD
?
c. In words, how does x relate to the measures of AB
Theorem: The measure of an interior angle formed by 2 intersecting chords is equal to
3. Consider the diagram to the right.
 = 70° and m∠AEB = 45°, find mCD
.
a. If mAB
 = 3 i mCD
 , find
b. If m∠AEC = 130° and mAB
.
mAB
10
Problems 4 and 5 consider the relationship between the angle formed by secants that intersect outside of a
circle and the measures of their intercepted arcs.
4. Consider the diagram. Draw auxiliary line BC .
a. Find m∠CBD.
b. Find m∠BCA.
c. Find x.
5. Repeat problem 4 with
 = 90° and mCD
 = 40°
a. mAB
 = w and mCD
 = z.
b. mAB
 and CD
?
c. In words, how does x relate to the measures of AB
Theorem: An exterior angle formed by 2 secants is equal to
6. Show that the same relationship holds when a tangent intersects a secant outside of a circle. Namely,
1
show that x = ( w − z ) .
2
Theorem: An exterior angle formed by a secant and a tangent is equal to
11
7. Consider the angle formed by two tangents in the diagram below. Draw BC .
a. Suppose that z = 40°. Find m∠ACB and
m∠ABC.
b. Find x.
c. What is x in terms of z? Why?
Theorem: An exterior angle formed by 2 tangents is
1.
2.
8. In the diagram below AC and AB are tangents. Show that x = y − 180 .
 : AB
 : BC
 = 5 : 6 : 7. Find the
9. A circle is inscribed in a triangle below. The arcs are in the ratio AC
measure of each angle of the triangle.
12
10. AB and AC are tangents below. If m∠A = 60° and m∠FDE = 105°,
.
a. Find mBC
.
b. Find mFE
11. AB and AC are tangent to the circle below. m∠E =42° and m∠BDC = 66°. Find m∠A.
 = 58°, mHG
 = 160°, and mEF
 = 2imCD
 . Find x.
12. Below mAB
Selected Answers to Classwork:
9) 40, 60, 80
10) (a) 120 (b) 90
11) 72 Hint: Let arc BC = x and arc GF = y
12) 63 Hint: let arc CD = y
End
13
Unit 7 Quiz Review (Day 5)
There are LOTS of formulas to know in this chapter! Study them first BEFORE you do this quiz review.
Write a 2-column proof for #1-3.
1. To the right, AB and CD are tangent to O .
 = mCY

Given: mBX
Prove: AB = CD
2. O is the center of the circle to the right.
Given: OC ⊥ AB
 = mBC

Prove: mAC
14
3. AC is the diameter of O .
a. Given: m∠3 = m∠4
Prove: AC bisects ∠DCB
b. Given: DC = BC
 = mAB

Prove: mAD
c. Given: m∠1 = m∠5
 = mAB

Prove: mAD
15
4. Find the value of x in each circle below.
(a)
(b)
(c)
(d)
(e)
(f)
5. Find the values of x and y in each circle below.
(a)
(b)
(c)
(d)
16
6. Find the values of x and y.
7. Find the value(s) of x.
8. In the diagram to the right, the circle is inscribed in ΔABC. AB = 14, BC = 10, and AC = 0. Find BD.
9. In the diagram to the right, AB is tangent to circle O. Find the radius of the circle.
17
10. In the diagram to the right, both AB and CD are tangent to O . OB = 29, AB = 21, and ED = 5.
Find CD.
11. Circles D, E, F, and O are tangent where they meet. Circles E and F have the same radius. If the radius
of circle D is 8, find AB.
Selected Answers to Quiz Review:
4) (a) 40 (b) 70 (c) 130 (d) 40 (e) 129 (f) 105
5) (a) x = 60, y = 30 (b) x = 123, y = 237 (c) x = 70, y = 25
6) x = 95, y = 55 7) x = 10 8) BD = 7.5
9) r = 8
(d) x = 100, y = 20
10) CD = 15
11) AB = 48
End
18
6-5 The Lengths of Chords (Day 7)
In this classwork, we will consider the relationship between chords and other segments within a circle.
1. Find the lengths of x and y in the circles below.
(a)
(b)
x
x
110
4
O
y
90
Theorem: Equal chords
y
Theorem: Chords equidistant from the center
of a circle
2. O is the center of the circle below. D is the midpoint of AB . AO = 29 and CD = 8.
a. Find OD.
b. Find AB.
3. A chord of length 24 is in a circle with area 169π. Find the distance of the chord to the center of the
circle. (DYOD)
19
4. Consider the circle below. In this problem, you will establish the relationship that, in the circle
shown, 8x = 4i6 .
a. Draw in AD and BC .
 = 100°, what are m∠A and m∠B?
b. Recall: If mDC
c. Show that ΔDAE ~ ΔCBE.
d. Show that 8x = 4i6
e. Solve for x.
Theorem: When two chords intersect in a circle,
5. Using the theorem above, find the value of x in each circle below.
(a)
(b)
x
10
2
5
x
4
10
8
(c)
(d)
x+1
8
x
4
12
x+6
x
x
9
x
y
20
1
2
6. In the circle below, E is the midpoint of BC . If AF = 2x + 2, EB =
x
, and AE = 2, find x.
3
7. The circle below has radius 10. If m∠A = 60°, find BC. (Hint: 30-60-90)
 = 2imAB
 , find the area
8. A rectangle ABCD is inscribed in a circle with circumference 12π. If mAD
of the rectangle. (DYOD)
9. In a circle, two parallel chords of length 4 and 8 are 6 units apart. Find the distance from each chord
to the center of the circle.
Selected Classwork Answers:
5) (c) x = 2 or 3 (d) x = 6, y = 18
6) x = 36 7) BC =
8)
9) 2 and 4
End
21
6-6 The Lengths of Tangents and Secants (Day 8)
In this classwork, we will consider the relationship between line segments formed by secants and tangents
that intersect outside the circle.
1. Consider the diagram with secant segments AC and EC .
a. Draw in AD and EB .
b. Why does ∠A = ∠E?
c. Show that ΔCAD ~ ΔCEB.
d. Use an appropriate proportion to
verify that CBiCA = CDiCE .
e. Suppose that AB = 6, BC = 4, and EC = 8. Find CD and ED.
Theorem: When two secants intersect at a common exterior point,
2. Use the theorem above to find each of the following:
(a)
(c) Two secants are drawn to a circle from an
(b)
5
15
6
4
x
10
25
external point. One secant has length 8 and
its internal segment has length 5. The length
of the internal segment of the other secant is
five times the length of its external segment.
Find the length of the second secant.
x
22
3. Consider the diagram with tangent segment AB and secant segment BC .
a. Draw AC and AD .
b. Show that ΔBDA ~ ΔBAC.
c. Show that BDiBC = AB 2
Theorem: When a secant and a tangent intersect at a common exterior point,
4. (a) Find x.
x
(b) Find the radius, r.
3
9
8
12
23
5. In the super fun circle below, IC : ID = 5 : 4. Also, HB = 3HF, and BE = 2HF.
a. Find IC and ID.
b. Find HF and HB.
c. Find EB.
d. Find ED.
6. In the diagram below, AD is tangent to the circle. Note that it is NOT given that AB passes through
the center. Find x and y.
Selected Classwork Answers:
2) (a) 4 (b) 20 (c) 12
4) (a) 6 (b) 5
5) (a) 15 and 12 (b) 4 and 12 (c) 8 (d) 5
6) y = 2, x = 4
End
24
6-7 Circumference and Area (Day 9)
Circumference of a Circle =
Area of a Circle =
Arc Length of a circle =
Area of a Sector =

1. (a) Find the length of PQ
if r = 3 and central ∠POQ = 110°
(b) Find the area of sector POQ.
O
P
Q
2. Find the circumference of a circle whose area of 121π.
3. In the circle to the right, m∠AOB = 40°. The shaded
.
has area 9π square units. Find the length of AB
region
4. The circles below have radius 2. They are tangent to each other and to AB . Find the area of the
shaded region.
25
5. A square is inscribed in a circle that is inscribed in a
square. The large square has perimeter 24. Find the area of
the shaded region.
6. A circle is inscribed in an equilateral triangle. The triangle is then inscribed in a circle. (DYOD) Find
the following ratios.
Area of small circle
a.
Area of triangle
b.
Area of small circle
Area of big circle
7. Find the area of the shaded region in circle O.
60°
O
12
 =60°. Find the shaded region.Error!
8. Each circle has radius 6 and mAB
26
9. The 5 small circles to the right are congruent and tangent where they
meet. Their centers lie on the two perpendicular segments. If the shaded
area is 100π, find the circumference of the large circle.
10. A circle is inscribed in an isosceles triangle with base 14. If the triangle has area 168, find the area of
the circle. (DYOD)
11. The three circles to the right have radius 4. Each circle is tangent to the two others. Find the shaded
area.
Selected Classwork Answers:
7) 24
–36
8) 12
-18
9) 30
10)
11) 16
–8
End
27
Unit 7 Circles Opportunity Review (Day 10)
There are LOTS of formulas to know in this chapter! Know them all well before you do this review.
1. Find x.
3.
2. Find x and y.
 =42° and mCD
 =92°, find m∠ADC.
AB  DC . If mBC
4. In the diagram below PA is tangent to the circle at A.
Points B, C, and P are collinear. If m∠BDC = 82° and
m∠CAP = 40°, find m∠ACP.
28
5. Find the ratio
unshaded area
for the equally spaced concentric circles below.
shaded area
6. A circle is inscribed in a square of side 10. AB is drawn parallel to one side of the square, one unit
away from that side. Find AX.
 = 96°, mCD
 = 30°, and mDE
 =20°.
7. In the diagram below, HG is tangent to the circle at B, mBC
Find the measures of the angles 1 through 6.
29
8. AB and EC are tangent to each circle below. If AB : EC = 4 : 3, and (CD)(ED)=50, find AE.
9. SR is tangent to circles P and Q. QT = 12, TR = 16, and PR = 45. Find PS, PQ, and ST.
 = 2imUT
,
10. In Q , it is given that US ⊥ RT , mRS
and RT = 24.
.
a. Find mUT
b. Find QV and QU.
30
11. In the diagram AB = AC and DB and DC are tangents. If m∠BDC = 84°, find x, y, and z.
12. AB and AC are tangent to the circle. If m∠E = 58° and m∠BDC = 88°, find m∠A.
Answers to Opportunity Review:
1) 84
2) x = 4, y = 8
3) 113
4) 138
8) 15
9) 27, 25, 20
10) (a) 60 (b) 4
5)
and 8
6) 2
7) 90, 63, 25, 57, 90, and 32
11) x = 66, y = 132, z = 48
12) 34
End
31
EXTRA SUPER DUPER FUN CHALLENGES:
YIKES!
1. AB is a diameter of semicircle with center O. Circles P and Q are tangent to each other and to the
semicircle. If OB = 4, find the radius of circle Q.
2. Circles of radii 3 and 6 are tangent to each other and to a circle of radius 9. AB is tangent to the two
smaller circles. Find AB.
Answers to the Extra Super Duper Fun Challenge Yikes Problems:
1) Answer: 1 (Big hint: Draw segment from P tangent to circle Q.)
2) 4
End
32