CAHSEE Geometry and Measurement Student Text - UC Davis - August 2008
CAHSEE Geometry and Measurement Student Text - UC Davis - August 2008
CAHSEE Geometry and Measurement Student Text - UC Davis - August 2008
CAHSEE on Target
Mathematics Curriculum
Published by
The University of California, Davis,
School/University Partnerships Program
2006
Director
Sarah R. Martinez, School/University Partnerships, UC Davis
Editor
Nadia Samii, UC Davis Nutrition Graduate
Reviewers
Faith Paul, School/University Partnerships, UC Davis
Linda Whent, School/University Partnerships, UC Davis
The CAHSEE stands for the California High School Exit Exam. The
mathematics section of the CAHSEE consists of 80 multiple-choice
questions that cover 53 standards across 6 strands. These strands
include the following:
Each student will receive a separate workbook for each strand and will
use these workbooks during their tutoring sessions. These workbooks
will present and explain each concept covered on the CAHSEE, and
introduce new or alternative approaches to solving math problems.
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Student Workbook: Geometry & Measurement
1. Pick the attribute you want to measure (such as the length, the
width, the perimeter, the area, the volume, the weight, the surface
area).
Example: Maurice would like to know how long his desk is. This
involves the following steps:
1. Pick the attribute of the desk to measure: length
2. Choose the unit of measure: combination of feet & inches
3. Measure the length of the desk: Use a ruler to find the
number of feet and inches.
On Your Own:
Evelyn would like to know how heavy her puppy is. She must….
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6 feet ● 12 inches
1 foot
6 ● 12 inches = 72 inches
Eric is 72 inches tall.
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Conversion Factors
* Means that you will need to memorize these conversions for the
CAHSEE. The others will probably be given to you.
You should also know the standard abbreviations and symbols for
units of measure:
Note: You should also know standard conversion factors for time:
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Steps:
20 ft ● 12 in
1 1 ft
20 ● 12 in = 20 ● 12 in = ________
1 1 1
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In the above, problem, we could also have written the fraction (in the
third step) as follows:
Steps:
1 yard
3 feet ← Feet in Denominator
5
15 feet ● 1 yard ← We are really just multiplying 15 ft by 1!
1 13 feet
5 ● 1 yd = ____ yds
1
Answer: There are _____ yards in 15 feet.
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Steps:
Since we are given the unit of tons in the problem, we want tons
to appear in the denominator of the fraction so we can cancel out
common units:
Solve:
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On Your Own:
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3. Mr. Adam buys 18 quarts of milk each week for his 5 children.
How many gallons of milk does this equal?
4. Rita needs 6 cups of milk for a pie recipe. How many fluid ounces
of milk does this represent?
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Comparing Measures
Steps:
3 feet = 1 yard
3 feet
1 yard ← Yards in Denominator
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Notice that the measures are expressed in different units. The first is
expressed in feet ('), the second in inches ("), and the third in both
feet (') and inches (").
12" = 1'
5 ft ● 12 in
1 ft
5 ft ● 12 in ← We can cancel out common units.
1 ft
5 ● 12 in = ____ in
• To convert 3' 20" to inches, let's first convert the 3 feet and then
add the 20 inches at the end:
3 ft ● 12 in
1 ft
3 ft ● 12 in ← We can cancel out common units.
1 ft
3 ● 12 in = ____ in
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On Your Own:
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5 lb 45 oz 4 lb 6 oz
5 lb = ______ 45 oz 4 lb 6 oz = _______
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3. The table below shows the departure and arrival times for four
different trains from Sacramento to Santa Barbara.
Hint: Figure out the travel time for each train. Then compare.
5. Miriam spent 450 minutes studying for her final exam in algebra.
Express the time in hours and minutes.
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1. Maria rode her bike 1 14 miles. How many feet did she ride on her
bike? (5,280 feet = 1 mile)
A. 6,340
B. 6,600
C. 7,180
D. 7,392
1
A. of a meter
1000
1
B. of a meter
100
C. 100 meters
D. 1000 meters
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4. The table below shows the flight times from San Francisco (S.F.) to
New York (N.Y.).
5. A boy is two meters tall. About how tall is the boy in feet ( ft ) and
inches (in ) ? (1 meter ≈ 39 inches)
A. 5 ft 0 in.
B. 5 ft 6in.
C. 6 ft 0 in.
D. 6 ft 6 in.
6. Juanita exercised for one hour. How many seconds did Juanita
exercise?
A. 60
B. 120
C. 360
D. 3,600
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Note: This means that for every 1 inch on the scale drawing, there
are 10 yards of actual classroom space.
There are two methods that we can use to solve this problem:
• With Algebra
• Without Algebra
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A. Length
Note: The units of the numerators must be the same, and the
units of the denominators must be the same. In the example
above, we will set the unit of the numerator as inches and the unit
of the denominator as yards:
inches inches
=
yards yards
1 inch = 2 inches
10 yards x yards
1x = 20 ← Cross multiply
x = 20 The actual length of the room is 20 yards.
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Student Workbook: Geometry & Measurement
B. Width
Let's look at this problem again and solve for the actual width of the
classroom:
Width:
Let y = the actual width of the classroom
1 inch = 1.5 inches
10 yards y yards
1y = ___ ← Cross multiply
y = ___ The actual width of the room is ___ yards.
We can also solve this problem without algebra. Just apply the ratio
of the actual size to the scale size.
Length
Scale: 1 scale inch represents 10 yards
Width
Scale: 1 scale inch represents 10 yards
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1 6 scale scale
= ← Set up your proportion: =
18 x actual actual
x = ____ ← Cross multiply
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On Your Own:
The drawing below represents an actual figure. If the scale is 1 inch =
18 inches, find the width of the actual figure.
With Algebra:
Without Algebra:
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A. 6 cm
B. 24 cm
C. 36 cm
D. 54 cm
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Unit 3: Rates
Example: A car that travels 60 miles in one hour travels at the rate of
60 miles per hour.
5mi
First, let's write this as a ratio:
60 min
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12 15 8 6
60 60 24 54
• With Algebra
• Without Algebra
1 = x ← Miles
5 50 ← Minutes
5x = 50 ← Cross multiply
5x = 50 ← Divide both sides by 5 to isolate the x.
5 5
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• Find the rate (by reducing the given ratio to lowest terms:
8 1
= Rate: 1 mile per 5 minutes
40 5
• Multiply the rate (in fraction form) by the quantity we are given in
the problem (50 minutes).
1 mi_ ● 50 min
5 min
10
1 mi_ ● 50 min ← Cancel out common units and factors!
15 min
Solve: 1 mi ● 10 = ___
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2. Irma ran 18 miles at the speed of 4½ miles per hour. How long did
it take her to run that distance?
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Comparing Rates
Example: Tim ran 150 meters in 25 seconds, and Evan ran 90 meters
in 15 seconds. Based on these rates, which statement is true?
A. Tim’s average speed was 4 meters per second faster than Evan’s
average speed.
B. Tim’s average speed was 2.4 meters per second faster than
Evan’s average speed.
C. Tim’s average speed was 2 meters per second faster than Evan’s
average speed.
D. Tim’s average speed was equal to Evan’s average speed.
To solve this problem, we need to find the rate in reduced form for
each runner. Once we have both rates in reduced form, we can
compare them:
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Ordering Rates
In this problem, it may be easiest to use the unit of feet, since two
rates are expressed in feet. Also, we may wish to use minutes, since
two rates are expressed in minutes.
We can now multiply the rate (35 yd/min) by the conversion factor
(expressed in fraction form) and cancel out common units:
continued →
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1 hour____
60 minutes
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1. Beverly ran six miles at the speed of four miles per hour. How long
did it take her to run that distance?
2
A. hr
3
1
B. 1 hrs
2
C. 4 hrs
D. 6 hrs
2. Sixty miles per hour is the same rate as which of the following?
A. 1 mile per minute
B. 1 mile per second
C. 6 miles per minute
D. 360 miles per second
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On the CAHSEE, you will be asked to find the perimeter (the distance
around the outside of an object or shape) and area (the size of the
inside of an object or shape) of both polygons and circles.
Think of two other situations where you would need to know the
perimeter.
• ______________________________________________________
• ______________________________________________________
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Note: Although 3 and 3.5 are shown only once in the figure, we need
to count them twice, since the sides opposite are of equal measure:
On Your Own:
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5 + __ + 2 + __ = ___ feet
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Perimeter = ___ cm
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When would you ever need to measure area? List three situations
below that would require the measure of area:
• ______________________________________________________
• ______________________________________________________
• ______________________________________________________
• ______________________________________________________
i. Rectangles
A = bh
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On Your Own:
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Height Base
5 ft = ____ in 12 ft = ____ in
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Area = _______
Area = _______
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ii. Squares
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On Your Own:
2. Find the area of a square with a length of 8". Area = ___ in²
3. Find the area of a square whose base is 11'. Area = ___ ft²
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5. The wall hanging below measures 10' by 10'. Find the perimeter
and the area of the hanging.
Perimeter: ____
Area: _____
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iii. Triangles
Note: Since the sum of all angles in a triangle is equal to 180º, each
of the three angles in an equilateral triangle is equal to 1/3 of 180º, or
60º.
4. A right triangle has a right angle (an angle that measures 90º).
Note: The two sides that form the 90º angle are perpendicular.
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Area of a Triangle
1
Area of a Triangle = bh
2
Note: Base (b) and height (h) are the horizontal and vertical sides
of the triangle, respectively. The diagonal line is called the
hypotenuse and is not used to find the area of the triangle.
1
A= bh
2
1
A= (8)(6) = ___ square inches
2
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On Your Own:
1
A= bh
2
1
A= (__)(__) = ___ units²
2
Note: The diagonal line is the hypotenuse. Use only the base and
height to solve this problem.
A = ___________
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Now find the area of one of the right triangles. (Note: We use the
same height but half of the base.)
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iv. Circles
To find the area of a circle, just take the square of the radius (r) and
then multiply it by the number ∏ (pi), which rounds to 3.14.
A = ∏ ● r²
A = ∏ ● (4 cm)²
A = 3.14 ● 16 cm² = ___ cm²
Note: On the CAHSEE, answer choices may be given in terms of ∏.
In these problems, you will not need to multiply by 3.14. Just solve
for r² and write ∏ beside the answer.
For the example above, write the area in terms of ∏: A = ___∏ cm²
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Note that in this example, we are given the diameter, not the radius.
To find the area, we need to first figure out the radius.
A = ∏r²
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Practice:
3. Find the area of the smaller circle. Give your answer in terms of ∏.
A = ____________
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A = ___∏ in²
A = ___∏ cm²
6. In the circle below, r is equal to 9 cm. Find the area of the circle.
A = ___∏ cm²
A = ___∏ in²
A = ___∏ cm²
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Circumference
d = diameter
r = radius
∏ ≈ 3.14
Since we are given the diameter, let's use the first formula: C = ∏d
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On Your Own:
C = ∏(__) = ___ cm
C = _________
C = _________
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A. 44 square units
B. 60 square units
C. 88 square units
D. 120 square units.
2. Louis calculated the area of the circle below and got an answer of
50.769 cm2. He knew his answer was wrong because the correct
answer should be about:
A. 4 X 4 X 4 = 64
B. 3 X 3 X 40 = 360
C. 31 X 4 X 4 = 496
D. 3 X 40 X 40 = 4800
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Circle x Circle y
4. The width of the rectangle shown below is 6 inches (in.). The length
is 2 feet (ft).
6 in.
2 ft.
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5. In the figure below, the radius of the inscribed circle is 6 inches (in ) .
What is the perimeter of square ABCD?
A. 12∏ in.
B. 36∏ in.
C. 24 in.
D. 48 in.
6. The points (1, 1), (2, 3), (4, 3), and (5, 1) are the vertices of a
polygon. What type of polygon is formed by these points?
A. Triangle
B. Trapezoid
C. Parallelogram
D. Pentagon
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Cube Three-dimensional
solid whose bases
(faces) are squares
Cone Three-dimensional
triangular solid
whose base is a circle
Three-dimensional
Pyramid triangular solid
whose base is a
rectangle
Cylinder Three-dimensional
rectangular solid with
two congruent bases
that are circles
Sphere Three-dimensional
circular solid with all
points the same
distance from the
center
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On the CAHSEE, you will be asked to find the volume and surface
area of solid figures. Solid figures are three-dimensional. Unlike
two-dimensional figures, which are flat, three-dimensional
figures have depth.
A. Volume
Let's look at the various types of prisms that may appear on the
CAHSEE.
i. Rectangular Prisms
Example
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V = 6 ● 3 ● 5 = ___
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On Your Own:
V = ___ cm3
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V = ___ units3
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ii. Cubes
Volume of a Cube
To find the volume of a cube, you can apply the formula for finding the
volume of a rectangular prism (V = lwh) or, since all of the bases are
equal, you can just cube one of the sides:
Volume of a Cube = s3
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On Your Own:
V = ___ units3
V = ________________
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Enlarging Cubes
On the CAHSEE, you may be asked to find the volume of a cube that
has been enlarged by a certain number or multiplied by a certain
number. Look at the following example:
Example: If the sides of the cube below are multiplied by two, the
volume of the new cube is ____.
A. Doubled
B. Multiplied by 3
C. Multiplied by 6
D. Multiplied by 8
V = 23 = 2 ● 2 ● 2 = 8
• If we multiply all of the sides by two, each side of the enlarged cube
will be 4 units (2 ● 2).
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V = 43 = 4 ● 4 ● 4 = 64
Answer: The volume of the new cube (64) is ____ times greater
than that of the first cube (8). Therefore, our answer is Choice ___.
On Your Own:
1. The sides of the cube below are multiplied by 4. Find the volume of
the new cube.
Should we . . .
A. Double the height?
B. Double both the length and width?
C. Double both the length and height?
D. Double the length, the width and the height.
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iii. Cylinders
A cylinder is a solid in which the bases are circles and the other
surface is a rectangle wrapped around the circle. (Note: Most drink
cans are cylinders.)
V = r²∏h
Note: It is a good idea to learn the formulas for the volume of a cube,
rectangular prism, and cylinder, although, in the past, these formulas
have been provided on the CAHSEE.
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On Your Own:
V = r²∏h
V = ____∏ in3
V = ___∏ mm3
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B. Surface Area
Note: Since each face has an identical match on the other side of the
prism; we can save time by multiplying each separate area by 2:
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On Your Own
As we have learned, to find the surface area of a solid, find the area
of each face and add them all together. However, since a cube is
made of 6 equal squares, we only need to find the area of one
square (s²) and multiply this number by 6.
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SA = 6s²
On Your Own
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Practice
SA = _______________
SA = ___________________
SA = __________________
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1. In the figure below, an edge of the larger cube is 3 times the edge
of the smaller cube. What is the ratio of the surface area of the
smaller cube to that of the larger cube?
A. 1 : 3
B. 1 : 9
C. 1 : 12
D. 1: 27
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A. 60π in.³
B. 120π in.³
C. 300π in.³
D. 600π in.³
A. 29
B. 75
C. 510
D. 675
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If Gina paints only the outside of the tool chest, what is the total
surface area, in square inches (in.²), she will paint?
A. 368
B. 648
C. 1296
D. 2880
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Note: The lengths of some of the sides are not written. You must
figure them out!
One way to make sure that you don’t forget a side in your calculations
is to put a dot in the upper left corner, and then go clockwise,
adding sides as you go around:
13 + __ + __ + __ + __ + __ = __ inches
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On Your Own:
1. Find the perimeter of the figure below:
P = __ + __+ __ + __ + __ + __ + __ + __ = _______
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We can see that this figure consists of two smaller regular figures:
One square
One rectangle
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On the CAHSEE, you may also be asked to find the remaining area of
a figure or the area of a shaded part of a figure:
• We can see that this irregular figure consists of two smaller regular
figures:
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On Your Own
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• The first step is to find the number of cubes. (Don’t forget to count
the cubes that you can't see but know are there!) ____
V = s3
V = (__) 3
V = ___
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A. 40 in.²
B. 44 in.²
C. 48 in.²
D. 52 in.²
A. 2,100
B. 2,800
C. 21,000
D. 28,000
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3. In the figure shown below, all the corners form right angles. What
is the area of the figure in square units?
A. 67
B. 73
C. 78
D. 91
A. 19 in.²
B. 29 in.²
C. 32 in.²
D. 38 in.²
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A. 20
B. 30
C. 50
D. 80
6. What is the area of the shaded region in the figure shown below?
(Area of a triangle = ½bh)
A. 4 cm²
B. 6 cm²
C. 8 cm²
D. 16 cm²
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A. Congruence
Objects that are exactly the same size and shape are said to be
congruent. This means that the corresponding sides (sides that
match up) will have the same length and that the corresponding
angles (angles that match up) will be equal in degrees.
Figures that have the same shape are called similar figures. They
may be different sizes; they may even be turned (rotated). However,
as long as the shape remains the same, they are similar.
Note: They have the same shape (so they are similar), but they do
not have the same size (so they are not congruent)
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CAHSEE Alert!
Congruent Triangles
M A
S R
A
K
B C
Q
C
L T V D
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≅ means congruent
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On Your Own:
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______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
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B. Geometric Transformations
i. Translations
Notice how the size and shape of the figure remains the same. All
that changes is the position.
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Example: The triangle on the left side of the graph has been
translated 3 units down and 5 units to the right.
Notice that the size and shape of the figure remain the same. All
that changes is its position.
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On Your Own:
1. The triangle on the left has been translated ___ units to the right.
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Student Workbook: Geometry & Measurement
Which graph best represents the graph of this parabola that has been
translated 2 units up?
A B
C D
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Student Workbook: Geometry & Measurement
ii. Reflections
To understand reflections, think about what happens when you look in
the mirror. You see the exact image of yourself, but reversed.
What happens when you look in the mirror and raise your right hand?
Notice that in the mirror image, the left hand appears to be raised.
What you're seeing in the mirror is a reflection. A reflection is the
image of a geometric figure that has been flipped over a line of
reflection (or a line of symmetry). We call reflections "flips" because
the figure flips over the line of reflection.
Line of Reflection
Line of reflection
Note: The lines above are also lines of symmetry since, in each case,
the line breaks the figure into two symmetric (equal and congruent)
parts.
Examples of a Reflection:
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Student Workbook: Geometry & Measurement
On Your Own:
Answer: _____________
Answer: ____________
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Student Workbook: Geometry & Measurement
Answer: ___________
Answer: _____________
Answer: ________________
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Student Workbook: Geometry & Measurement
6. Which figure will result if the triangle is reflected across the y-axis?
A B
C D
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Student Workbook: Geometry & Measurement
8.
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Student Workbook: Geometry & Measurement
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Student Workbook: Geometry & Measurement
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Student Workbook: Geometry & Measurement
a² + b² = c² where . . .
If we know any two sides of a right triangle, we can find the third
by plugging the known values into the formula.
Find the value of b: ___ The second length has a length of ___"
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Student Workbook: Geometry & Measurement
Example: Find the height of the right triangle (formed by dividing the
equilateral triangle in half).
a² + b² = c²
Note: The base of the equilateral triangle is 16. What is the base
of the right triangle? ___
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Student Workbook: Geometry & Measurement
Pythagorean Triples
There are three sets of Pythagorean triples that appear over and
over in math problems. Knowing them will save you a lot of time.
Every set of multiples will also give the sides of a right triangle.
If you memorize the first set of triples, you can easily find the
multiples. Just keep multiplying by the same factor.
a b c
3 4 5
6 8 10
9 12 15
12 16 20
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a b c
5 12 13
a b c
8 15 17
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Student Workbook: Geometry & Measurement
On Your Own:
b = ________
a = _________
c = ________
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b = __________
a = ___________
x = ___________
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y = __________
8. Two points in the x-y-plane have coordinates (1, 5) and (3, 1). The
distance between them is equal to the square root of which
number?
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Student Workbook: Geometry & Measurement
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