A tangent line is always perpendicular to the radius drawn to the point of tangency.. Just think of a bicycle tire the circle on the road the tangent: notice that the center of the wheel
Trang 1Circle Basics
Okay, we all know a circle when we see one, but it
often helps to know the mathematical definition of a
circle
• A circle is all of the points in a plane that
are a certain distance r from the center.
• The radius is the distance from the center
to any point on the circle
Radius means ray in Latin; a radius comes from
the center of the circle like a ray of light from the
sun.
• The diameter is twice the radius: d = 2r.
Dia- means through in Latin, so the diameter is
a segment that goes all the way through the circle.
The Circumference and Area
It’s easy to confuse the circumference formula with
the area formula, because both formulas contain the
same symbols arranged differently: circumference=
2πr and area = πr2 There are two simple ways to avoid
that mistake:
• Remember that the formulas for
circumfer-ence and area are given in the refercircumfer-ence
in-formation at the beginning of every math
section
• Remember that area is always measured in
square units, so the area formula is the one
with the “square:” area = πr2
Tangents
Lesson 8: Circles
diameterradius
tangent
radius
A tangent is a line that touches (or intersects) the
cir-cle at only one point Think of a plate balancing on its
side on a table: the table is like a tangent line to the plate
A tangent line is always perpendicular to the radius drawn to the point of tangency
Just think of a bicycle tire (the circle) on the road (the tangent): notice that the center of the wheel must be
“directly above” where the tire touches the road, so the radius and tangent must be perpendicular
M
P
R
l
M
P
R
l
7 5
Example:
In the diagram above, point M is 7 units away from the center of circle P If line l is tangent to the circle and MR= 5, what is the area of the circle?
First, connect the dots Draw MP and PR to make a
triangle
Since PR is a radius and MR is a tangent, they are
perpendicular
Since you know two sides of a right triangle, you can use the Pythagorean theorem to find the third side: 52+ (PR)2= 72
Simplify: 25 + (PR)2= 49 Subtract 25: (PR)2= 24
(PR)2is the radius squared Since the area of the circle is πr2, it is 24π
Trang 21 What is the formula for the circumference of a circle?
2 What is the formula for the area of a circle?
3 What is a tangent line?
4 What is the relationship between a tangent to a circle and the radius to the point of tangency?
5 In the figure above, AB –– is a tangent to circle C, AB = 8, and AD = 6 What is the circumference of circle C?
6 In the figure above, P and N are the centers of the circles and are 6 centimeters apart What is the area of
the shaded region?
Concept Review 8: Circles
C A
D
B
8
r
Trang 31. Two circles, A and B, lie in the same plane If
the center of circle B lies on circle A, then in
how many points could circle A and circle B
intersect?
I 0
II 1
III 2
(A) I only
(B) III only
(C) I and III only
(D) II and III only
(E) I, II, and III
2. What is the area, in square centimeters, of a circle
with a circumference of 16π centimeters?
(A) 8π
(B) 16π
(C) 32π
(D) 64π
(E) 256π
3. Point B lies 10 units from point A, which is the
center of the circle of radius 6 If a tangent line
is drawn from B to the circle, what is the
dis-tance from B to the point of tangency?
Note: Figure not drawn to scale
4. In the figure above, AB ––– and AD ––– are tangents
to circle C What is the value of m?
Note: Figure not drawn to scale
5. In the figure above, circle A intersects circle B
in exactly one point, is tangent to both
cir-cles, circle A has a radius of 2, and circle B has
a radius of 8 What is the length of CD –––?
CDs ruu
C
D
C A
D
B
SAT Practice 8: Circles
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1 0 2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6
1
2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6
1
2 3 4 5 7 8 6
1 0 2 3 4 5 7 8 6
1 0 2 3 4 5 7 8 6
1 0 2 3 4 5 7 8 6
Trang 4Concept Review 8
1 circumference = 2πr
2 area = πr2
3 A tangent line is a line that intersects a circle at
only one point
4 Any tangent to a circle is perpendicular to the
radius drawn to the point of tangency
Answer Key 8: Circles
62+ x2= 102 Simplify: 36 + x2= 100
Take the square root: x= 8
4 45 Since AB ––– and AD ––– are tangents to the circle, they are perpendicular to their respective radii, as shown The sum of the angles in a quadrilateral is
360°, so
m + 3m + 90 + 90 = 360
Simplify: 4m+ 180 = 360 Subtract 180: 4m= 180
C A
D
B
SAT Practice 8
5 Draw BC ––– to make a right triangle, and call the
length of the radius r Then you can use the
Pythagorean theorem to find r:
82+ r2= (r + 6)2 FOIL: 64 + r2= r2+ 12r + 36
Subtract r2: 64 = 12r + 36
Subtract 36: 28 = 12r
Divide by 12: 7/3 = r
The circumference is 2πr, which is 2π(7/3) = 14π/3.
C A
D
B
8
r
6 Draw the segments shown
here Since PN is a
ra-dius of both circles, the radii of both circles have the same length Notice
that PN, PR, RN, PT, and
NT are all radii, so they are
all the same length; thus, ΔPNT
and ΔPRN are equilateral triangles and their
angles are all 60° Now you can find the area of the left half of the shaded region
This is the area of the sector minus the area of ΔRNT Since
∠RNT is 120°, the sector is
120/360, or 1⁄3, of the circle The circle has area 36π, so the sec-tor has area 12π ΔRNT consists
of two 30°-60°-90° triangles, with sides as marked, so its area is
Therefore, half of the origi-nal shaded region is , and the whole is
24π −18 3
12π −9 3
1 2 6 3 3/ 9 3
( ) ( ) ( )=
R
T S
N
R
T
6
6
3 √3 3√3 3
B
B
A
1 E The figure above demonstrates all three
possibilities
2 D The circumference = 2πr = 16π Dividing by 2π
gives r = 8 Area = πr2= π(8)2= 64π
3 8 Draw a figure as shown, including the tangent
segment and the radius extended to the point of
tangency You can find x with the Pythagorean
theorem:
A
B
6 10
x
Trang 55 8 Draw the segments shown Choose point E to
make rectangle ACDE and right triangle AEB
No-tice that CD = AE, because opposite sides of a
rec-tangle are equal You can find AE with the
Pythagorean theorem:
(AE)2+ 62= 102 Simplify: (AE)2+ 36 = 100
Take the square root: AE= 8
Since CD = AE, CD = 8.
C
A
B
D
2
2 6
E
Trang 6ESSENTIAL ALGEBRA 2 SKILLS
CHAPTER 11
✓ Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use
Trang 7Analyzing Sequences
A sequence is just a list of numbers, each of which is
called a term An SAT math question might ask you to
use a sequential pattern to solve a problem, such as
“How many odd numbers are in the first 100 terms of
the sequence 1, 2, 3, 1, 2, 3, ?”
An SAT sequence question usually gives you
the first few terms of a sequence or the rule for
generating the sequence, and then asks you
ei-ther to find a specific term in the sequence (as
in “What is the 59th term of this sequence?”)
or to analyze a subset of the sequence (as in
“What is the sum of the first 36 terms of this
se-quence?”) To tackle sequence problems:
1 Use the pattern or rule to write out the first
six to eight terms of the sequence
2 Try to identify the pattern in the sequence
Notice in particular when the sequence
be-gins to repeat itself, if it does
3 Use this pattern, together with
whole-num-ber division (Chapter 7, Lesson 7), if it’s
helpful, to solve the problem
Example:
1, 2, 2, The first three terms of a sequence are shown above
Each term after the second term is found by dividing
the preceding term by the term before that For
exam-ple, the third term is found by dividing the second
term, 2, by the first term, 1 What is the value of the
218th term of this sequence?
Don’t panic You won’t have to write out 218 terms! Just write out the first eight or so until you notice that the sequence begins to repeat The fourth term is
2 ÷ 2 = 1, the fifth term is 1 ÷ 2 = 1/2, and so
on This gives the sequence 1, 2, 2, 1, 1/2, 1/2,
1, 2, Notice that the first two terms of the sequence,
1 and 2, have come back again! This means that the first six terms in the sequence, the underlined ones, will just repeat over and over again Therefore,
in the first 218 terms, this six-term pattern will repeat
times, or 36 with a remainder of 2 So, the 218th term will be the same as the second term in the sequence, which is 2
Example:
1, 1, 0, 1, 1, 0,
If the sequence above repeats as shown, what is the sum of the first 43 terms of this sequence?
Since the sequence clearly repeats every three terms, then in 43 terms this pattern will repeat 43 ÷ 3
= 14 (with remainder 1) times Each full repetition of the pattern 1, 1, 0 has a sum of 0, so the first 14 rep-etitions have a sum of 0 This accounts for the sum of the first 14 3 = 42 terms But you can’t forget the
“remainder” term! Since that 43rd term is 1, the sum of the first 43 terms is 1
You won’t need to use the formulas for “arith-metic sequences” or “geometric sequences” that you may have learned in algebra class Instead, SAT “sequence” questions simply require that you figure out the pattern in the sequence
218 6 362
3
÷ =
Lesson 1: Sequences
There’s a lot of detail to learn and understand to do well on the SAT For more tools and resources that will help, visit our Online Practice Plus at www.MHPracticePlus.com/SATmath.
Trang 8Concept Review 1: Sequences
1 What is a sequence?
2 If the pattern of a sequence repeats every six terms, how do you determine the 115th term of the sequence?
3 If the pattern of a number sequence repeats every four terms, how do you find the sum of the first 32 terms
of the sequence?
4 If a number sequence repeats every five terms, how do you determine how many of the first 36 terms are negative?
5 What is the 30th term of the following sequence?
, 1, 3, 9,
6 The first term in a sequence is 4, and each subsequent term is eight more than twice the preceding term What is the value of the sixth term?
7 The word SCORE is written 200 times in a row on a piece of paper How many of the first 143 letters are vowels?
8 The third term of a sequence is x If each term in the sequence except the first is found by subtracting 3 from the previous term and dividing that difference by 2, what is the first term of the sequence in terms of x?
9 A 60-digit number is created by writing all the positive integers in succession beginning with 1 What is the 44th digit of the number?
1
9,
1
3
Trang 9SAT Practice 1: Sequences
1. The first term in a sequence is x Each
subse-quent term is 3 less than twice the preceding
term What is the fifth term in the sequence?
(A) 8x 21
(B) 8x 15
(C) 16x 39
(D) 16x 45
(E) 32x 93
1/8, 1/4, 1/2,
2. In the sequence above, each term after the
first is equal to the previous term times a
con-stant What is the value of the 13th term?
(A) 27
(B) 28
(C) 29
(D) 210
(E) 211
3. The first term in a sequence is 400 Every
sub-sequent term is 20 less than half of the
imme-diately preceding term What is the fourth
term in the sequence?
4. In the number 0.148285, the digits 148285 re-peat indefinitely How many of the first 500 digits after the decimal point are odd? (A) 83
(B) 166 (C) 167 (D) 168 (E) 332
5, 6, 5, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 5,
5. In the sequence above, the first 5 is followed by one 6, the second 5 is followed by two 6s, and so
on If the sequence continues in this manner, how many 6s are there between the 44th and 47th appearances of the number 5?
(A) 91 (B) 135 (C) 138 (D) 182 (E) 230
6. The first term in a sequence is 5, and each subsequent term is 6 more than the immedi-ately preceding term What is the value of the 104th term?
(A) 607 (B) 613 (C) 618 (D) 619 (E) 625
7. What is the units digit of 336? (A) 0
(B) 1 (C) 3 (D) 7 (E) 9
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0
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5
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8
9
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1 0 2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6
Trang 10210, 70,
8. After the first term in the sequence above, each
odd-numbered term can be found by
multiply-ing the precedmultiply-ing term by three, and each
even-numbered term can be found by
multi-plying the previous term by 1⁄3 What is the
value of the 24th term?
9. The first two terms of a sequence are 640 and
160 Each term after the first is equal to
one-fourth of the previous term What is the value
of the sixth term?
6, 4,
10. After the first two terms in the sequence above, each odd-numbered term can be found by divid-ing the previous term by 2 For example, the third term is equal to 4 ÷ 2 = 2 Each even-numbered term can be found by adding 8 to the previous term For example, the fourth term is equal to 2 + 8 = 10 How many terms are there before the first noninteger term?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
2, 4, 8,
11. The first three terms of a sequence are given above If each subsequent term is the product
of the preceding two terms, how many of the first 90 terms are negative?
(A) 16 (B) 30 (C) 45 (D) 60 (E) 66
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1 0 2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6