Download Geometry

Objective 12.1A
New Vocabulary
plane
plane figures
space
solids
line
line segment
parallel lines
intersecting lines
ray
angle
vertex
degree
right angle
perpendicular lines
complementary angles
straight angle
supplementary angles
acute angle
obtuse angle
New Symbols
|| (is parallel to)
∠ (angle)
° (degrees)
∟ (right angle)

(is perpendicular to)
Discuss the Concepts
1. Describe each of the following: ray, line, and line segment.
2. Does the surface of Earth lie in a plane?
3. Using any objects in the classroom, provide examples of each of the following: a right angle, an acute
angle, an obtuse angle, a plane, intersecting lines, parallel lines, and perpendicular lines. (If students
need some assistance, suggest that they consider the windows.)
Concept Check
1. Provide three names for the angle below.
∠O, ∠AOB, ∠BOA
2a. Give the number of degrees in a full circle. 360°
b. Give the number of degrees in a straight angle. 180°
c. Give the number of degrees in a right angle. 90°
3. How many dimensions does each of the following have?
a. A point 0
b. A line 1
c. A line segment 1
d. A ray 1
e. An angle 2
4. What is the name given to lines in a plane that do not intersect? Parallel lines
5. What is the name given to two lines that intersect at right angles? Perpendicular lines
Optional Student Activity
1. Prepare a handout on which you have drawn four angles. You might include a 40° angle, a 60° angle, a
120° angle, and a 30° angle. Orient the angles in different directions. Make enough copies of the
handout so that each student in the class will have one.
During class, provide students with protractors and instruction on how to measure angles. Give each
student a copy of the handout. Have them measure each angle and then classify each angle as an
acute angle or an obtuse angle.
2. On a number line, the points A, B, C, and D have coordinates −2.5, 2, 5, and 3.5, respectively. Which of
these points is halfway between two others? D, or 3.5
3. Find the measure of the smaller angle between the hands of a clock when the time is 5 o’clock. 150°
4. On a line, Q is between P and S. R is between Q and S. S is between Q and T. PT = 28 units, QS = 8
units, and PQ = QR = RS. Find ST. 16 units
Objective 12.1B
New Vocabulary
triangle
base of a triangle
height of a triangle
right triangle
hypotenuse
legs of a right triangle
quadrilateral
parallelogram
height of a parallelogram
rectangle
square
circle
center of a circle
diameter of a circle
radius of a circle
geometric solid
rectangular solid
cube
sphere
center of a sphere
diameter of a sphere
radius of a sphere
cylinder
New Formulas
The angles in a triangle: ∠A + ∠B + ∠C = 180°
r
1
2
d
d = 2r
Discuss the Concepts
1. In a right triangle, why are the two acute angles complementary?
2. Which of the following are plane figures: a square, a cube, a triangle, a sphere, a circle, a point, and a
box?
Concept Check
1. The shape of Earth approximates what geometric figure? A sphere
2. The angles of a triangle are in the ratio 2:3:7. Find the number of degrees in the largest angle. 105°
Optional Student Activity
If time permits, you might teach your students how to prepare a circle graph. If you did the first activity
described on page 517, your students know how to measure angles. You will need to provide them with
instructions on drawing angles.
We suggest that you provide your students with papers on which you have drawn a few large circles for the
graphs. The center of each circle should be marked with a dot. Have them prepare circle graphs for the
data provided in the exercises that follow.
a. Shown below are American adults’ favorite pizza toppings. (Source: Market Facts for Bolla wines)
Pepperoni
43%
Sausage
19%
Mushrooms
14%
Vegetables
13%
Others
7%
Onions
4%
b. According to a Pathfinder Research Group survey, more than 94% of adults have heard of the Three
Stooges. The choices among those who have a favorite are as follows:
Curly
52%
Moe
31%
Larry
12%
Curly Joe 3%
Shemp
2%
Objective 12.1C
Vocabulary to Review
intersecting lines [12.1A]
parallel lines [12.1A]
supplementary angles [12.1A]
New Vocabulary
vertical angles
adjacent angles
transversal
alternate interior angles
alternate exterior angles
corresponding angles
Discuss the Concepts
What is a transversal? Describe two different ways in which a transversal can intersect two other lines.
Concept Check
When a transversal intersects two parallel lines, which of the following are supplementary angles: vertical
angles, adjacent angles, alternate interior angles, alternate exterior angles, or corresponding angles?
Adjacent angles
Answers to Writing Exercises
67. ∠AOC and ∠BOC are supplementary angles. Therefore, ∠AOC + ∠BOC = 180°. Because ∠AOC =
∠BOC, by substitution ∠AOC + ∠AOC = 180°. Therefore, 2(∠AOC) = 180° and ∠AOC = 90°. Therefore,
AB is perpendicular to CD.
Objective 12.2A
Vocabulary to Review
plane geometric figure [12.1A]
triangle [12.1B]
right triangle [12.1B]
quadrilateral [12.1B]
parallelogram [12.1B]
rectangle [12.1B]
square [12.1B]
New Vocabulary
polygon
sides of a polygon
regular polygon
isosceles triangle
equilateral triangle
scalene triangle
acute triangle
obtuse triangle
perimeter
circumference
New Symbols
π (pi)
New Formulas
Perimeter of a triangle: P = a + b + c
Perimeter of a rectangle: P = 2L + 2W
Perimeter of a square: P = 4s
Circumference of a circle: C = πd or C = 2πr
Discuss the Concepts
1. Is every square a rectangle? Is every rectangle a square?
2. In the definition of a polygon, what does the phrase closed figure mean? Draw a figure that is not closed.
Concept Check
1. Figure A below is a rectangle. Label the length of the rectangle L and the width of the rectangle W.
Figure A
2. What is the name of a regular polygon that has three sides? Equilateral triangle
3. What is the name of a parallelogram in which all angles are the same measure? Rectangle
Optional Student Activity
1. The base of isosceles triangle ABC (with AB = BC and ∠B = 42°) and the base of equilateral triangle
CDE lie on line segment AE. Find the measure of ∠BCD. 51°
B
A
D
C
E
2. Three line segments are randomly chosen from line segments whose lengths are 1 cm, 2 cm, 3 cm, 4
cm, and 5 cm. What is the probability that a triangle can be formed from the line segments?
3
10
3.
Triangle FJH is an isosceles triangle in which FJ = FH, FK = KJ, FG = GH. Find the perimeter of
triangle FJH if FK = 2x + 3, GH = 5x − 9, and JH = 4x. 60 units
F
G
H
K
J
Objective 12.2B
New Vocabulary
composite geometric figure
Optional Student Activity
(This activity is adapted from James Gray Propp’s “The Slicing Game,” American Mathematical Monthly,
April 1996. Used with permission.)
Shown below is a triangle with three “slices” through it. Note that the resulting pieces are diamonds,
trapezoids, and a triangle.
Determine how you can slice a triangle so that all the resulting pieces are triangles. Use three slices. Each
slice must cut all the way through the triangle.
Solution:
(Note: You might also ask students to determine how to slice (a) a quadrilateral, (b) a pentagon, (c) a
hexagon, and (d) a heptagon so that all the resulting pieces are triangles.)
Objective 12.2C
Optional Student Activity
1. In triangle ABC, AB = 10 ft. BC = (x + 7) ft. and AC = (2x + 3) ft. The perimeter of the triangle is 32 ft. Is
triangle ABC scalene, isosceles, or equilateral? Isosceles
2. The lengths of two sides of a triangle are 10 in. and 14 in. What are the possible values for x, the length
of the third side of the triangle? x is greater than 4 in. and less than 24 in.
Answers to Writing Exercises
43. The ranger could measure the circumference of the trunk of the tree and then solve the equation C =
πd for d.
Objective 12.3A
New Vocabulary
area
New Symbols
in2 (square inches)
cm2 (square centimeters)
ft2 (square feet)
m2 (square meters)
mi2 (square miles)
New Formulas
A
1
bh
2
Area of a triangle:
Area of a rectangle: A = LW
Area of a square: A = s2
Area of a circle: A = πr2
Discuss the Concepts
What is wrong with the statement?
a. The perimeter is 40 m2.
b. The area is 120 ft.
Concept Check
The concepts of square units and area are difficult for students. After introducing these ideas, ask students
questions such as the following:
1. Would I measure the distance from Chicago to Boston in miles or square miles? Miles
2. Is the amount of land cultivated by a gardener measured in feet or square feet? Square feet
3. How is the size of a state park measured? Acres or square miles
4. How is the length of a bedroom measured? Feet or meters
Optional Student Activity
1. Use graph paper to draw different rectangles, each with a perimeter of 20 units. Investigate just wholenumber dimensions. What dimensions will result in a rectangle with the greatest possible area? 5 x 5
8 
 d
9 
2
2. Ancient Egyptians gave the formula for the area of a circle as
, where d is the diameter. Does
this formula give an area that is less than or greater than the area given by the correct formula? Greater
than
3. Choose a local pizza restaurant and a particular type of pizza. Determine the size (by diameter) and cost
of a medium pizza and of a large pizza. (Use regular prices with no special discounts.) Find the ratio of
the cost of a medium pizza to the size of a medium pizza and the ratio of the cost of a large pizza to the
size of a large pizza. Explain what each ratio represents. Are the ratios approximately equal? Explain
your findings and what they mean.
Objective 12.3B
Vocabulary to Review
composite geometric figure [12.2B]
Optional Student Activity
Geometric probability involves using lengths or areas of geometric figures to determine the likelihood that
an event will occur. Here are two exercises involving geometric probability.
1. Floating in a swimming pool measuring 8 m by 4.5 m is an inflatable raft measuring 0.8 m by 2.5 m. If an
autumn leaf lands on a random point in the pool, what is the probability that it will land on the raft?
1
18
2. Gary Timken has a bulletin board that measures 16 in. by 24 in. in his college dorm room. On that
bulletin board is a photo that measures 3.5 in. by 5.5 in. If a dart is thrown and lands at a random point
on the bulletin board, what is the probability that the dart will hit the photograph? Write the answer as a
decimal rounded to the nearest hundredth. 0.05
Objective 12.3C
Optional Student Activity
(Note: For the first activity below, students will need to be able to look at an American flag.)
1. An American flag has dimensions of 65 in. by 39 in. Each short stripe has a length of 39 in. What
fractional part of the area of the flag is red?
27
65
2. A rectangular box that has no top is made from pieces of cardboard. If the box is 8 in. long, 7 in. wide,
and 6 in. high, what is the total area of the cardboard used to construct the box? 236 in2
3. What is the maximum number of 3” x 5” index cards that will fit completely and without overlap on an
8.5” x 11” piece of paper? 5 index cards
Answers to Writing Exercises
42. The area in the circles to the left of the line is equal to the area in the circles to the right of the line.
Note that in the circle at the left in the top row, the line goes through the center of the circle; thus it is a
diameter of the circle, and half the area lies on one side of the line and half lies on the other side of the
line. A complete circle lies on each side of the line; the circle at the right in the top row is on one side,
and the circle at the left in the bottom row lies on the other side. For the two circles at the right on the
bottom row, half their combined area lies on the left side of the line, and half lies on the right side of the
line.
Objective 12.4A
New Vocabulary
volume
New Symbols
in3 (cubic inches)
cm3 (cubic centimeters)
ft3 (cubic feet)
m3 (cubic meters)
New Formulas
Volume of a rectangular solid: V = LWH
Volume of a cube: V = s3
Volume of a sphere:
Volume of a cylinder:
V 
4
 r3
3
V   r 2h
Discuss the Concepts
The difficulty students have distinguishing linear measure from square measure is compounded with
volume measure. Ask students to give examples of things that would be measured in, for instance, feet,
square feet, and cubic feet—for example, the length of a room, the area of the floor, and the volume of air in
the room. Here are two more examples.
a. The distance across a lake, the area of the surface of the lake, and the volume of water in the lake
b. The length of a driveway, the area of the driveway that needs to be plowed, and the volume of asphalt
used to pave the driveway
Concept Check
1. Indicate which of the following are rectangular solids: a juice box, a milk carton, a can of soup, a
compact disk, the plastic container a compact disk is packaged in. A juice box and the plastic container
a compact disk is packaged in
2. Indicate which of the following units could not be used to measure the volume of a cylinder: ft 3, m3, yd2,
cm3, mi. Yd2, mi
Optional Student Activity
1. A foot is what fraction of a yard?
1
3
2. A square foot is what fraction of a square yard?
3. A cubic foot is what fraction of a cubic yard?
4. An inch is what fraction of a foot?
1
9
1
27
1
12
5. A square inch is what fraction of a square foot?
6. A cubic inch is what fraction of a cubic foot?
7. A centimeter is what fraction of a meter?
1
144
1
1728
1
100
8. A square centimeter is what fraction of a square meter?
9. A cubic centimeter is what fraction of a cubic meter?
1
10, 000
1
1, 000, 000
Objective 12.4B
New Vocabulary
composite geometric solid
Optional Student Activity
The dimensions of a grocery bag are 7 in. by 12 in. by 17 in. The dimensions of a lunch bag are 4 in. by 6
in. by 12 in.
a. Find the difference between the volume of a grocery bag and a lunch bag. 1140 in3
b. If 3 in. is added to the height of a lunch bag, what is the change in the volume? 72 in3
c. How much larger is the volume of a lunch bag if 3 in. is added to the width of the base of the bag
rather than to its height? 144 in3
d. If 3 in. is added to each dimension of a lunch bag, what is the change in the volume? 657 in3
e. Is this more or less than twice the original volume? More than twice the original volume
Optional Student Activity
(Note: You might ask students to estimate an answer to Exercise 2 below before performing the
calculations. They will probably be surprised at the result.)
1. When an object is placed in water, the object displaces an amount of water that is equal to the volume of
the object.
a. A sphere with a diameter of 4 in. is placed in a rectangular tank of water. Find the volume of the
water that is displaced by the sphere. Round to the nearest tenth. 33.5 in3
b. The radius of the base of a cylinder is 2 cm, and its height is 10 cm. The cylinder is submerged in
a tank of water. Find the volume of the water that is displaced by the cylinder. Round to the
nearest tenth. 125.6 cm3
2. A cylindrical oil tank has a diameter of 100 ft and a height of 70 ft. How many gallons of oil can be stored
in the tank? 7.48 gal of oil can be stored in a volume of 1 ft3. Round to the nearest thousand. 4,110,000
gal
Answers to Writing Exercises
38. For example, cut perpendicular to the top and bottom faces and parallel to two of the sides.
39. For example, beginning at an edge that is perpendicular to the bottom face, cut at an angle through to
the bottom face.
40. For example, beginning at the top face, at a distance d from a vertex, cut at an angle to the bottom
face, ending at a distance greater than d from the vertex directly below the first chosen vertex.
41. For example, beginning on the top face, at a distance d from a vertex, cut across the cube to a point
just below the opposite vertex, intersecting the bottom face.
42. The length of the rectangular solid is equal to half the circumference of the base of the cylinder.
L
1
2
C
The width of the rectangular solid is equal to the radius of the base of the cylinder.
W=r
The height of the rectangular solid is equal to the height of the cylinder.
H=h
For the base of the rectangular solid:
L
1
2
C; W  r
A  LW 
1
2
C (r ) 
1
2
1
( d ) r   (2r ) r   r 2
2
The volume of the rectangular solid and the cylinder is V = LWH = (LW)h = πr2h.
Objective 12.5A
New Vocabulary
square root
perfect square
New Symbols
(square root)
Discuss the Concepts
What is a perfect square? How can you mentally find the square root of a perfect square?
Optional Student Activity
1. Find a perfect square that is between 350 and 400. 361
2. What is the smallest integer larger than
3 5?4
3. If x = 4, what is the value of 3 x  8 ? 2
4. Find the two-digit perfect square that has exactly nine factors. 36
5. Find two whole numbers such that their difference is 10, the smaller number is a perfect square, and the
larger number is two less than a perfect square. 14 and 4
6. Find the smallest integer n such that
1 2  3 
 n simplifies to an integer. 8
Objective 12.5B
Vocabulary to Review
hypotenuse [12.1B]
legs [12.1B]
New Vocabulary
Pythagorean Theorem
New Formulas
Hypotenuse 
Leg 
 leg    leg 
2
2
 hypotenuse    leg 
2
2
Discuss the Concepts
How can you tell which is the hypotenuse and which are the legs of a right triangle?
Concept Check
1. Label the right triangle below. Include the right-angle symbol, the hypotenuse, and the legs.
2. The diagonal of a rectangle is a line drawn from one vertex to the opposite vertex. Find the length of the
diagonal of a rectangle that is 5 m wide and 11 m long. Round to the nearest tenth. 12.1 m
Optional Student Activity
The numbers 3, 4, and 5 are called a Pythagorean triple because they are natural numbers that satisfy the
equation of the Pythagorean Theorem.
In each exercise below, determine whether the numbers are a Pythagorean triple.
a. 5, 7, and 9 No
b. 8, 15, and 17 Yes
c. 11, 60, and 61 Yes
d. 28, 45, and 53 Yes
Mathematicians have investigated Pythagorean triples and have found formulas that will generate these
triples. One such formula is
a = m2 − n2
b = 2mn
c = m2 + n2, where m > n
For instance, let m = 2 and n = 1. Then a = 22 − 12 = 3, b = 2(2)(1) = 4, and c = 22 + 12 = 5. This is the
Pythagorean triple 3, 4, 5.
Find the Pythagorean triple produced by each of the following.
e. m = 3 and n = 1 6, 8, 10
f. m = 5 and n = 2 20, 21, 29
g. m = 4 and n = 2 12, 16, 20
h. m = 6 and n = 1 12, 35, 37
Objective 12.5C
Concept Check
(Note: For these exercises, encourage students to draw a diagram.)
1. A pilot flies 90 mi east from Landown Airport to Mitchell Airfield and then 60 mi north from Mitchell
Airfield to Wallace Airport. Find the distance of the return flight straight back to Landown from Wallace.
Round to the nearest mile. 108 mi
2. Marta Lightfoot leaves a dock in her sailboat and sails 2.5 mi due east. She then tacks and sails 4 mi
due north. The two-way radio Marta has on board has a range of 5 mi. Will she be able to call a friend
on the dock from her location using the two-way radio? Yes
Optional Student Activity
The radius of the circle shown below is 3 in. Find the length of a side of the square drawn inside the circle.
Round to the nearest tenth. 4.2 in.
Answers to Writing Exercises
42. To determine if a 25-foot ladder is long enough to reach 24 ft up the side of the home when the bottom
of the ladder is 6 ft from the base of the side of the house, use the Pythagorean Theorem to find the
hypotenuse of a right triangle with legs that measure 24 ft and 6 ft.
c 2 = a2 + b2
c2 = 242 + 62
c2 = 576 + 36
c2 = 612
c ≈ 24.74
Compare the leg of the hypotenuse with 25. If the hypotenuse is shorter than 25 ft, the ladder will reach
the gutter. If the hypotenuse is longer than 25 feet, the ladder will not reach the gutter.
24.74 < 25
The hypotenuse is shorter than 25 ft. The ladder will reach the gutters.
43. No. The Pythagorean Theorem applies to right triangles, and the triangle shown is not a right triangle.
Objective 12.6A
Vocabulary to Review
ratio [4.1A]
New Vocabulary
similar objects
similar triangles
congruent objects
congruent triangles
New Rules
side-side-side rule (SSS)
side-angle-side rule (SAS)
Discuss the Concepts
1. What is the difference between similar objects and congruent objects?
2. The text discusses two rules to determine whether two triangles are congruent, the SSS rule and the
SAS rule. Are two triangles congruent if all three angles of one triangle are equal to all three angles of
the second triangle?
3. What does the phrase “the included angle” mean?
Concept Check
1. Is it possible for two circles not to be similar? No
2. Is it possible for two circles not to be congruent? Yes
3. Use the SSS rule to draw two congruent triangles. Check that students have labeled all three sides of
each triangle.
4. Use the SAS rule to draw two congruent triangles. Check that students have labeled two sides and the
included angle on each triangle.
5. Draw two triangles, each with a 90° angle and sides that measure 3 in. and 4 in., that are not congruent.
Students might draw one triangle in which the hypotenuse is 5 in. and the legs measure 3 in. and 4 in.
and another in which the hypotenuse is 4 in. and the legs measure 3 in. and approximately 2.65 in.
Optional Student Activity
1. A pancake 4 in. in diameter contains 5 g of fat. How many grams of fat are in a pancake 6 in. in
diameter? Explain how you arrived at your answer. 11.25 g
2. In the figure below, triangles ADB, CDA, and BDC are congruent triangles. Find the measure of ∠DAC.
30°
A
C
D
B
Objective 12.6B
Optional Student Activity
1. The lengths of the three sides of a triangle are 3 m, 4 m, and 6 m. Find the least possible perimeter of a
similar triangle, one of whose sides measures 12 m. 26 m
Answers to Writing Exercises
20. Yes. Given two squares, the ratios of corresponding sides are equal because the same number will be
in the numerators (the length of a side of one square) and the same number will be in the
denominators (the length of a side of the second square.)
No. The lengths of the sides of a rectangle vary. Therefore, given two rectangles, the ratios of
corresponding sides may vary.
22. To find the height of the tree, you could use similar triangles and a proportion. This method assumes
that the sun is shining and is not directly overhead. Use the yardstick to measure the shadow of the
tree. Then use the yardstick to measure the length of your own shadow. Write and solve a proportion.
One possible proportion is as follows:
“Your height” over “the length of your shadow” is equal to “the height of the tree” over “the length of the
shadow cast by the tree.”
Answers to Focus on Problem Solving—Trial and Error
No, it is not possible for two different pairs of factors of a number to have the same sum.
Answers to Projects and Group Activities—Investigating Perimeter
1. 12 units
2. 10 units
3. 14 units
4. 10 units
Answers to Projects and Group Activities—Symmetry
1. No
2. Besides A and H, the capital letters B, C, D, E, I, M, O, T, V, W, X, and Y have axes of symmetry. Note
that the letters I, O, and X have more than one axis of symmetry. (For some letters, it depends on how
the letter is written; an example is the letter U.)
3. The lowercase letters c, i, l, t, v, and w have one axis of symmetry. (For some letters, it depends on how
the letter is written; an example is the letter t.)
4. The lowercase letters o and x have more than one axis of symmetry.
5. An isosceles triangle has one axis of symmetry. An equilateral triangle has three axes of symmetry. A
rectangle has two axes of symmetry. A square has four axes of symmetry. A circle has an infinite
number of axes of symmetry. A trapezoid has no axis of symmetry.
6. If a figure is unchanged after being rotated 180° about a point O, then the figure has point symmetry. A
parallelogram, a rectangle, a square, and a circle have point symmetry.
If a figure is unchanged after being rotated more than 0° and less than 360°, then the figure has
rotational symmetry. An equilateral triangle, a rectangle, a square, a circle, and a parallelogram have
rotational symmetry. It may be helpful to note that a regular pentagon has rotational symmetry but not
point symmetry.
7. Examples of symmetry in nature include people and animals, snowflakes, starfish, and many types of
flowers. Examples of symmetry in art and architecture can be found in history texts and in art and
architectural magazines.