Download 1-1 The Coordinate Plane pg. 6

The Coordinate Plane
TEKS/TAKS:
1.a, 2.b, 4.a, 7.a
Objective:
You will graph ordered pairs on a coordinate plane.
Used in:
Locating items in publishing, archaeology, and aquatic explorations.
Vocabulary:
Coordinate plane, x-axis, y-axis, origin, quadrants, ordered pair, xcoordinate, y-coordinate
Additional Reading:
Textbook 1-1 pg. 6
Real World Application
Aquatic Engineering
The Dutch Delta Plan, which controls the flow of the Atlantic
Ocean on the southwest coast of the Netherlands, is one of
the greatest achievements of engineering. The system was
built using a grid system of nylon mattresses with graded
gravel and rocks to support concrete piers and steel gates.
The horizontal axis is labeled with letters and the vertical axis
with numbers. Suppose the first five piers were placed at 5B,
2K, 8D, 7A, and 12E. Sketch a graph showing the positions
of the first five piers.
Challenge Problem
Make a table of values to determine the five points that lie on
the graph of y= x2-2x+6.
Use the following values of x: 1, 2, 3, 4, and 5.
Plot the five points.
Do they appear to be collinear?
Why or why not?
Challenge Homework
Pgs. 10-11
#27, 29, 34,
37, 47
Points, Lines, and Planes
TEKS/TAKS:
1.a, 1.b, 2.b, 4.a
Objective:
You will identify and model points, lines, planes, coplanar points,
intersecting lines and planes, and solve problems by listing
possibilities
Used in:
Representing real-life (tangible) objects
Vocabulary:
Planes, lines, points, space, possibilities
Additional Reading:
Textbook 1-2 pg. 12
Real World Application
Anatomy
Has anyone ever told you to stand up straight? If your
posture is perfect, you should be able to draw a straight line
from your ear to your ankle, running through your shoulder,
hip, and knee. Study the posture of five of your friends or
relatives. How many of them seem to have good posture
according to the straight line rule? What percent of the
people you observed have good posture?
Challenge Problem
The Hawaiian game of lu-lu is played with four
disks of volcanic stone. The face of each stone is
marked with a series of dots. A player tosses the
four disks and if they land all face-up, 10 points
are scored, and the player tosses again. If any of
the disks land facedown on the first toss, the
players gets to toss those pieces again. The score
is the total number of dots showing after the
second toss. List the possible outcomes after the
first toss.
Challenge Homework
Pgs. 16-18
#27, 35, 49,
57, 69
Measuring Segments
TEKS/TAKS:
1.a, 1.b, 2.a, 2.b, 4.a, 7.a, 7.c, 8.c
Objective:
You will find the distance between two points on a number line and
between two points in a coordinate plane and use the Pythagorean
Theorem to find the length of the hypotenuse of a right triangle.
Used in:
Segment measures are used in discovering characteristics of many
geometric shapes.
Vocabulary:
Between, measure, Ruler Postulate, Pythagorean Theorem, distance
formula
Additional Reading:
Textbook 1-4 pg. 28
Real World Application
World Records
On September 8, 1989, the British Royal Marines stretched a
rope from the top of Blackpool Tower (416 feet high) in
Lancashire, Great Britain, to a fixed point on the ground 1128
feet from the base of the tower. Then Sgt. Alan Heward and
Cpl. Mick Heap of the Royal Marines, John Herbert of
Blackpool Tower, and TV show hosts Cheryl Baker and Roy
Castle slid down the rope establishing the greatest distance
recorded in a rope slide. Draw a right triangle to represent
this event. How far did they slide?
Challenge Problem
Draw a figure that satisfies all of the following
conditions:
Points A, B, C, D, and E are collinear.
Point A lies between points D and E.
Point C is next to point A, and BD = DC.
Challenge Homework
Pgs. 33-35
#37, 39, 45,
47, 57
Midpoints and Segment
Congruence
TEKS/TAKS:
1.a, 2.a, 2.b, 4.a, 7.a, 7.c
Objective:
You will find the midpoint of a segment, and complete proofs
involving segment theorems.
Used in:
Finding the midpoint is often used to connect algebra to geometry.
Vocabulary:
Midpoint, segment bisector, theorems, proof, paragraph proof,
informal proof
Additional Reading:
Textbook 1-5 pg. 36
Real World Application
Transportation
Interstate 70 passes through Kansas. Mile markers are used
to name many of the exits. The exit for U.S. Route 283 North
is Exit 128, and the exit to use U.S. Route 281 to Russell is
Exit 184. The exit for Hays on U.S. Route 183 is 3 miles
farther than halfway between Exits 128 and 184. What is the
exit number for the Hays exit?
Challenge Problem
Point C lies on AB such that AC = ¼ AB. If the
endpoints of AB are A(8, 12) and B (-4, 0), find the
coordinates of C.
Challenge Homework
Pgs. 41-43
#35, 37, 41,
45, 55
Exploring Angles
TEKS/TAKS:
1.a, 1.b, 2.a, 2.b, 4.a, 5.a
Objective:
You will identify and classify angles, use the Angle Addition
Postulate, find the measures of angles, and identify and use
congruent angles and the bisector of an angle.
Used in:
Astronomy, construction, art, and engineering
Vocabulary:
Angle, opposite rays, sides, vertex, interior, exterior, degrees,
measure, Protractor Postulate, congruent, angle bisector
Additional Reading:
Textbook 1-6 pg. 44
Real World Application
Entertainment
John Trudeau used angles to help him design The Flintstones
pinball machine. The angle that the machine tilts, or the
pitch of the machine, determines the difficult of the game.
The angles at which the flipper will hit the ball are considered
when the ramps, loops, and targets of the game are placed.
Mr. Trudeau recommends that The Flintstones machine be
installed with a pitch of 6° to 7°. Is this an acute, right,
straight, or obtuse angle?
Challenge Problem
Draw BA and BC such that they are opposite rays
and BE bisects <ABD. If m<ABE = 6x + 2 and
m<DBE = 8x – 14, find m<ABE.
Challenge Homework
Pgs. 49-51
#15, 31, 37,
45, 49
Angle Relationships
TEKS/TAKS:
1.a, 2.a, 2.b, 4.a, 9.a
Objective:
You will identify and use adjacent, vertical, complementary,
supplementary, and linear pairs of angles, and perpendicular lines,
and to determine what information can and cannot be assumed from
a diagram.
Used in:
Geology, sports, and construction
Vocabulary:
Perpendicular lines and adjacent, vertical, supplementary, and
complementary angles, linear pair.
Additional Reading:
Textbook 1-7 pg. 53
Real World Application
Sports
In the 1994 Winter Olympic Games, Espen Bredesen of
Norway claimed the gold medal in the 90-meter ski jump.
When a skier completes a jump, he or she tries to make the
angle between his or her body and the front of his or her skis
as small as possible. If Espen is aligned so that the front of
his skis make a 15° angle with his body, what angle is formed
by the tail of the skis and his body?
Challenge Problem
<R and <S are complementary angles, and <U and
<V are also complementary angles.
If m<R = y – 2, m<S = 2x + 3, m<U = 2x – y,
and m<V = x – 1, find the values of x, y, m<R,
m<S, m<U, and m<V.
Challenge Homework
Pgs. 49-51
#13, 21, 25,
31, 45
Inductive Reasoning
and Conjecturing
TEKS/TAKS:
1.a, 2.b, 3.d, 4.a
Objective:
You will make conjectures based on inductive reasoning.
Used in:
Law, higher level mathematics and science, research
Vocabulary:
Inductive reasoning, conjecture, counterexample
Additional Reading:
Textbook 2-1 pg. 70
Real World Application
Billiards
Consider a carom billiard table with a length of 6
feet and a width of 3 feet. Suppose you start in
the upper left-hand corner and shoot the ball at a
45° angle. Use graph paper to trace the path of
the ball. Make a conjecture about shooting the ball
from any corner of a table this size at a 45° angle.
Challenge Problem
Determine if the conjecture below is true or false.
Explain your answer and give a counterexample if
it is a false conjecture.
Given: x is an integer.
Conjecture: -x is negative.
Challenge Homework
Pgs. 72-75
#9, 19, 23, 33,
39
If-Then Statements
and Postulates
TEKS/TAKS:
1.a, 2.b, 3.a, 3.b, 3.c, 3.d, 4.a
Objective:
You will write statements in if-then form, you will write the converse,
inverse, and contrapositive and you will identify and use basic
postulates about points, lines, and planes.
Used in:
Understanding if-then statements helps determine the validity of
conclusions.
Vocabulary:
If-then statements, conditional statements, hypothesis, conclusion,
converse, inverse, contrapositive, negation, Venn diagram
Additional Reading:
Textbook 2-2 pg. 76
Real World Application
Biology
Use a Venn diagram to illustrate the following
conditional about the animal kingdom.
“If an animal is a butterfly, then it is an arthropod.”
Challenge Problem
Consider the conditional, “If two angles are
adjacent, they are not both acute.”
Write the converse of the contrapositive of the
inverse of the conditional.
Explain how the result is related to the original
conditional.
Challenge Homework
Pgs. 72-75
#49, 53, 55,
57, 63
Deductive Reasoning
TEKS/TAKS:
1.a, 2.b, 3.c
Objective:
You will use the Law of Detachment and the Law of Syllogism
in deductive reasoning and you will solve problems looking for
a pattern.
Used in:
You can use deductive reasoning to reach logical conclusions.
Vocabulary:
Law of Detachment, deductive reasoning, Law of Syllogism
Additional Reading:
Textbook 2-3 pg. 85
Real World Application
Airline Safety
The statement below is posted in airports throughout the U.S. Provide
information necessary to illustrate logical reasoning using the Law of
Detachment with this if-then statement.
Attention All Travelers
If any unknown person attempts to give you any items
including luggage to transport on your flight, do not accept it
and notify airline personnel immediately.
Challenge Problem
Use deductive reasoning laws to write a true
conclusion using all of the following three
statements. Explain all steps used to arrive at
your conclusion; remember, if a conditional is true
then its contrapositive is true.
If a person is baby, then the person is not logical.
If a person can manage a crocodile, then that
person is not despised.
If a person is not logical, then the person is
despised.
Challenge Homework
Pgs. 72-75
#21, 29, 35,
41, 47
Using Proof in Algebra
TEKS/TAKS:
1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a
Objective:
You will use properties of equality in algebraic and geometric
proofs.
Used in:
Forensic science, law, higher-level mathematics and science
Vocabulary:
Proof, two-column proof, properties of equality, properties of
segments
Additional Reading:
Textbook 2-4 pg. 92
Real World Application
Physics
Kinetic energy is the energy of motion. The
formula for kinetic energy is Ek = h · f + W,
where h represents the work function of the
material being used. Solve this formula for
f and justify each step.
Challenge Problem
What are some of the similarities and differences
between the Transitive Property of Equality and the
Transitive Property of Congruent Segments? Give
an example of each property using segments and
angles.
Challenge Homework
Pgs. 96-99
#21, 27, 29,
37, 41
Verifying Segment
Relationships
TEKS/TAKS:
1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a
Objective:
You will complete proofs involving segment theorems.
Used in:
Geography
Vocabulary:
Reflexive, symmetric, and transitive
Additional Reading:
Textbook 2-5 pg. 100
Real World Application
Measurement
Some rulers have centimeters on one edge and inches on the
other edge.
About how long in centimeters is a segment that is 6 inches
long?
Are the two segments congruent? Explain.
Challenge Problem
Draw and complete the proof.
Given:
PS is congruent to RQ.
M is the midpoint of PS.
M is the mipoint of RQ.
Prove:
PM is congruent to RM.
Challenge Homework
Pgs. 104-106
#21, 27, 29,
33, 35, 43
Verifying Angle
Relationships
TEKS/TAKS:
1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a
Objective:
You will complete proofs involving angle theorems.
Used in:
Art, nature, and architecture
Vocabulary:
Illusion
Additional Reading:
Textbook 2-6 pg. 107
Real World Application
Architecture
The Leaning Tower of Pisa in Italy makes an angle
with the ground of about 84° on one side. If you
look at the building as a ray and the ground as a
line, then the angles that the tower forms with the
ground form a linear pair. Find the measure of the
other angle that the tower makes with the ground.
Challenge Problem
Given <5 and <A are complementary.
<6 and <A are complementary.
m<5 = 2x + 2 and m<6 = x + 32.
Find the m<5 and m<6.
Challenge Homework
Pgs. 112-114
#33, 35, 39,
41, 51
Parallel Lines and
Transversals
TEKS/TAKS:
1.a, 2.b, 3.b, 3.d, 3.e, 4.a, 9.a
Objective:
You will solve problems by drawing a diagram, you will identify
relationships between two lines or between two planes, and you will
name angles formed by a pair of lines and a transversal.
Used in:
Architecture, agriculture, and air travel
Vocabulary:
Drawing a diagram, skew lines, transversal, interior, exterior,
alternate exterior, consecutive interior, corresponding
Additional Reading:
Textbook 3-1 pg. 124
Real World Application
Music
The word “parallel” is used in music to describe
songs moving consistently by the same intervals
such as harmony with parallel voices. Find at least
two additional uses of the word “parallel” in other
school subjects such as history, electronics,
computer science, or English.
Challenge Problem
Square dancing involves four couples. If each
member of the square shakes hands with every
other member of the square except his or her
partner before the dance begins, what is the total
number of handshakes?
Challenge Homework
Pgs. 127-129
#15, 35, 41,
45, 55
Angles and Parallel Lines
TEKS/TAKS:
2.a, 2.b, 3.d, 9.a
Objective:
You will use the properties of parallel lines to determine angle
measure.
Used in:
Construction and interior decorating
Vocabulary:
None
Additional Reading:
Textbook 3-2 pg. 131
Real World Application
Interior Decorating
Walls in houses are never perfectly vertical. To hang
wallpaper, a true vertical line must be established so the
pattern looks nice. The paperhanger uses a plumb line, which
is a piece of string with a weight at the bottom, to make the
vertical line for the first piece of wallpaper. How can she be
sure that all of the pieces of wallpaper are vertical if she does
not use the plumb line again?
Challenge Problem
In the figure below, explain why you can conclude
that <1 is congruent to <4, but you cannot state
that <3 is necessarily congruent to <2.
A
B
1
2
3
4
D
C
Challenge Homework
Pgs. 135-137
#33, 39, 47,
53, 59
Slopes of Lines
TEKS/TAKS:
1.a, 1.b, 2.b, 4.a, 7.a, 7.b, 9.a
Objective:
You will find the slopes of lines and use slope to identify
parallel and perpendicular lines.
Used in:
Finding the distance between two points.
Vocabulary:
Slope, if and only if
Additional Reading:
Textbook 3-3 pg. 138
Real World Application
Construction
According to the building code in Crystal Lake,
Illinois, the slope of a stairway cannot be steeper
than 0.88. The stairs in Li-Chih’s home measure
11 inches deep and 7 inches high. Do the stairs in
his home meet the code requirements? Explain
your answer.
Challenge Problem
A line contains the points at (-3, 6) and (1, 2).
Using slope, write a convincing argument that the
line intersects the x-axis at (3, 0). Graph the
points to verify your conclusion.
Challenge Homework
Pgs. 142-144
#35, 37, 39,
43, 51
Proving Lines Parallel
TEKS/TAKS:
1.a, 1.b, 2.b, 4.a, 7.a, 7.b, 9.a
Objective:
You will recognize angle conditions that produce parallel lines
and prove two lines are parallel based on given angle
relationships.
Used in:
Construction and physics
Vocabulary:
None
Additional Reading:
Textbook 3-4 pg. 146
Real World Application
Construction
Carpenters use parallel lines in creating walls for
construction projects. Copy the drawing below.
Label your drawing and describe three different
ways to guarantee that the wall studs are parallel.
Challenge Problem
Suppose lines a, b, and c lie in the same plane and
a||b and a||c.
Draw a figure showing lines a, b, and c.
Explain how you would prove that b||c.
Challenge Homework
Pgs. 151-152
#27, 31, 33,
41, 43
Parallels and Distance
TEKS/TAKS:
1.a, 2.a, 2.b, 4.a, 7.a, 7.b, 9.a
Objective:
You will recognize and use distance relationships among
points, lines, and planes.
Used in:
Finding the distance between points and lines and between
parallel lines and parallel planes.
Vocabulary:
Equidistant
Additional Reading:
Textbook 3-5 pg. 154
Real World Application
Construction
Dominique wants to install vertical boards
to strengthen the handrail structure on her
deck. How can she guarantee the vertical
boards will be parallel?
Challenge Problem
Find the distance between point P(6, -2) and the
graph of line k whose equation is y = 7.
Challenge Homework
Pgs. 159-160
#25, 33, 35,
37, 39
Spherical Geometry
TEKS/TAKS:
1.a, 2.a, 2.b, 4.a, 7.a, 7.b, 9.a
Objective:
You will identify points, lines, and planes in spherical geometry and
compare and contrast basic properties of plane and spherical
geometry.
Used in:
Understanding the relationship of locations on the surface of Earth.
Vocabulary:
Plane Euclidean geometry, spherical geometry, non-Euclidean
geometry
Additional Reading:
Textbook 3-6 pg. 163
Real World Application
Space Travel
According to Einstein’s Spherical Universe,
geometric properties of space are similar to those
on the surface of a sphere. Based on this model,
what conclusion follows about the path of a
spaceship along a straight line? Explain your
reasoning.
Challenge Problem
Explain why triangle ABC could not exist in plane
Euclidean geometry. Could triangle ABC exist in
non-Euclidean spherical geometry? Explain your
reasoning. Include a drawing.
C
A
B
Challenge Homework
Pgs. 168-169
#18, 25, 31,
33, 39
Classifying Triangles
TEKS/TAKS:
1.a, 1.b, 2.b, 4.a, 7.a, 7.c, 9.b
Objective:
You will identify the parts of triangles and classify triangles by
their parts.
Used in:
Architecture and crafts
Vocabulary:
Triangle, polygon, sides, vertices, acute, obtuse, right
triangle, equiangular, scalene, isosceles, equilateral
Additional Reading:
Textbook 4-1 pg. 180
Real World Application
Architecture
Consider a figure of the basic structure of the geodesic
dome. How many equilateral triangles are in the figure.
How many obtuse triangles are in the figure?
Challenge Problem
The Pythagorean Theorem states that in a right triangle, the
square of the measure of the hypotenuse is equal to the sum
of the squares of the measures of the legs. Use patty paper
to create several obtuse and acute triangles. Measure the
sides of each. Square the measure of each side. Compare
the square of the measure of the longest side to the sum of
the squares of the measures of the other two sides. Make a
conjecture about the relationship of the square of the
measure of the longest side compared to the sum of the
squares of the measures of the two shorter sides.
Challenge Homework
Pgs. 185-187
#41, 45, 49,
51, 59
Measuring Angles in Triangles
TEKS/TAKS:
1.a, 1.b, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b
Objective:
You will apply the Angle Sum Theorem and you will apply the
Exterior Angle Theorem.
Used in:
Construction, design, and astronomy
Vocabulary:
Exterior angle, remote interior angle, flow proof
Additional Reading:
Textbook 4-2 pg. 189
Real World Application
Astronomy
Leo is a constellation that represents a lion. Three of the brighter
stars in the constellation form a triangle LEO. If the angles have
measures indicated in the figure at the right, find m<L.
O
27°
L
Leo
E
93°
Challenge Problem
In triangle ABC, m<A is 16 more than
m<B, and m<C is 29 more than m<B.
Write an equation relating the
measures and find the measure of
each angle.
Challenge Homework
Pgs. 185-187
#29, 39, 41,
43, 45
Exploring Congruent Triangles
TEKS/TAKS:
1.a, 1.b, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.a, 10.b
Objective:
You will name and label corresponding parts of congruent
triangles.
Used in:
Crafts, arts, and construction
Vocabulary:
Congruent triangles, congruence transformation
Additional Reading:
Textbook 4-3 pg. 196
Real World Application
Crafts
Use graph paper to design a quilt using congruent
triangles. Classify the triangles used by name and
identify those that can be proved congruent.
Challenge Problem
On graph paper, draw six congruent right scalene
triangles. Cut out the triangles. Arrange the
triangles so that congruent sides fit together. Try
several different arrangements.
How many different shapes with four sides can you
make?
How many different shapes with three sides can
you make?
Challenge Homework
Pgs. 201-203
#25, 31, 41,
43, 51
Proving Triangles Congruent
TEKS/TAKS:
1.a, 1.b, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 7.a, 9.b, 10.a, 10.b
Objective:
You will use SSS, SAS, and ASA Postulates to test for triangle
congruence.
Used in:
Determine if two triangles are congruent
Vocabulary:
CPCTC
Additional Reading:
Textbook 4-4 pg. 206
Real World Application
Recreation
Tapatan is a game played in the Phillipines on a square board as shown at the right. The players take
turns placing each of their three pieces on a different point of intersection. After all the pieces have
been played, the players take turns moving a piece along a line to another intersection. A piece
cannot jump over another piece. A player who gets all his or her pieces in a straight line wins the
game. Point E bisects all four line segments that pass through it.
What can you say about triangle GHE
A
B
C
and triangle CBE? Explain.
What you can say about triangle AEG
and triangle IEGA? Explain.
What can you say about triangle ACI
D
F
E in the
middle
and triangle CAG? Explain.
G
H
I
Challenge Problem
Write a proof.
Given: <J is congruent to <L
B is the midpoint of JL.
Prove: triangle JHB is congruent to triangle LCB.
H
J
B
C
L
Challenge Homework
Pgs. 210-212
#23, 27, 35,
37, 39
More Congruent Triangles
TEKS/TAKS:
1.a, 1.b, 2.b, 3.b, 3.d, 3.e, 4.a, 9.b, 10.a, 10.b
Objective:
You will use the AAS Theorem to test triangle congruence,
and to solve problems by eliminating possibilities.
Used in:
Broadcasting
Vocabulary:
Eliminate the possibilities
Additional Reading:
Textbook 4-5 pg. 214
Real World Application
History
It is said that Thales determined the distance from the shore to
enemy Greek ships during an early war by sighting the angel to the
ship from a point P on the shore, walking a distance to point Q, and
then sighting the angle to the ship from that point. He then
reproduced the angles on the other side of line PQ and continued
these lines until they intersected. How did he determine the
distance to the ship in this way?
Why does this method work?
P
Q
Challenge Problem
Can two triangles be proved congruent by AAA
(Angle-Angle-Angle)?
Justify your answer completely.
Challenge Homework
Pgs. 218-221
#25, 37, 41,
43, 45
Analyzing Isosceles Triangles
TEKS/TAKS:
1.a, 2.b, 4.a, 9.b, 10.b
Objective:
You will use the properties of isosceles and equilateral
triangles.
Used in:
Design, carpentry, and navigation
Vocabulary:
None
Additional Reading:
Textbook 4-6 pg. 222
Real World Application
Navigation
Sau-Lim is the captain of a ship and he uses an instrument called a
“pelorus” to measure the angle between the ship’s path and the line
from the ship to a lighthouse. Sau-Lim finds the distance that the
ship travels and the change in the measure of the angle with the
lighthouse as the ship sails. When the angle with the lighthouse is
twice that of the original angle, Sau-Lim knows that the ship is as
far from the lighthouse as the ship has traveled since the lighthouse
was first sighted. Why?
Challenge Problem
Draw an isosceles triangle ABC with vertex angle at
A. Find the midpoints of each side. Label the
midpoint of AB point D, the midpoint BC point E,
and the midpoint of AC point F. Draw triangle DEF.
Name a pair of congruent triangles. Explain your
reasoning.
Name three isosceles triangles. Explain your
reasoning.
Challenge Homework
Pgs. 225-227
#23, 25, 29,
33, 47
Applying Congruent Triangles
TEKS/TAKS:
1.a, 2.b, 3.b, 3.c, 4.a, 7.a, 7.b, 9.b, 10.b
Objective:
You will identify and use medians, altitudes, angle bisectors,
and perpendicular bisectors in a triangle.
Used in:
Engineering, sports, and physics.
Vocabulary:
Perpendicular bisector, median, altitude, angle bisector
Additional Reading:
Textbook 5-1 pg. 238
Real World Application
Physics
Physicists often make calculations based on the center of gravity of an object. Follow
the steps and answer the questions below to investigate the center of gravity of a
triangle.
a. Draw a large acute triangle that is not isosceles.
b. Construct the three medians of the triangle. What do you notice?
c. Construct three angle bisectors of the triangle. What do you notice?
d. Construct three altitudes of the triangle. What do you notice?
e. Construct the three perpendicular bisectors of the sides of the triangle. What do you
notice?
f. Cut out the triangles you made for parts a-e. Place the point where the segments
intersect on the flat end of a pencil for each of the triangles. What do you observe?
g. What changes would occur in the construction in parts b-e if the triangle were right
or obtuse instead of acute?
Challenge Problem
Draw any triangle ABC with median AD and
altitude AE. Recall that the area of a
triangle is one-half the product of the
measures of the base and the altitude.
What conclusion can you make about the
relationship between the areas of triangle
ABD and triangle ACD?
Challenge Homework
Pgs. 243-244
#27, 33, 41,
45, 47
Right Triangles
TEKS/TAKS:
1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b
Objective:
You will recognize and use tests for congruence of right
triangles.
Used in:
Trigonometry
Vocabulary:
Leg, hypotenuse
Additional Reading:
Textbook 5-2 pg. 245
Real World Application
Construction
A wooden bridge is being constructed over a stream in a
park. The braces for the support posts came from the
lumberyard already cut. The carpenter measures from the
top of the support post to a point on the post to find where
to attach the brace to the post. Explain why only one
measurement must be made to ensure that all of the
braces will be in the same relative position.
Challenge Problem
In the figure below, m<W = m<X = m<Y =
45. XB | WY, YA | WX.
X
If WZ = 10, find XY.
A
Z
W
B
Y
Challenge Homework
Pgs. 249-251
#21, 23, 35,
37, 45
Indirect Proofs and
Inequalities
TEKS/TAKS:
1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b
Objective:
You will use indirect reasoning and indirect proof to reach a
conclusion, you will recognize and apply properties of inequalities to
the measures of segments and angles, and you will solve problems
by working backwards.
Used in:
Law and advertising
Vocabulary:
Indirect reasoning, indirect proof, working backward
Additional Reading:
Textbook 5-3 pg. 252
Real World Application
Law
The defense attorney said to the jury,
“My client is not guilty. According to
the police, the crime occurred on July
14 at 6:30 p.m. in Boston. I can
prove that at that time my client was
attending a business meeting in New
York City. A verdict of not guilty is
the only possible verdict.” Is this an
example of indirect reasoning?
Explain why or why not.
Challenge Problem
Mr. Mendez was checking on the date he
had attended a four-day conference a year
ago. The page in his record book was torn
and all that remained of the date for the
meeting was “ber 31”. What was the month
of the first day of the conference?
Challenge Homework
Pgs. 256-258
#27, 31, 37,
43, 47
The Triangle Inequality
TEKS/TAKS:
1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b
Objective:
You will apply the Triangle Inequality Theorem.
Used in:
Carpentry and mathematics history
Vocabulary:
Triangle Inequality Theorem
Additional Reading:
Textbook 5-5 pg. 267
Real World Application
Carpentry
Salina is building stairs and wants to nail a
brace at the base of each stair as shown in
the figure. The brace attaches at the bottom
of the rise and anywhere along the tread. The
stairs have an 18-cm rise and a 26-cm tread.
There is a pile of braces 5 centimeters, 20
centimeters, 24 centimeters, and 45
centimeters long that Salina can use. Which
of the lengths can she sue as a brace?
Challenge Problem
Is it true that the difference between any
two sides of a triangle is less than the third
side? Explain your reasoning?
Challenge Homework
Pgs. 270-272
#37, 45, 47,
53, 59
The Triangle Inequality
TEKS/TAKS:
1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b
Objective:
You will apply the SAS Inequality and the SSS Inequality.
Used in:
Physical therapy and biology
Vocabulary:
SAS, SSS
Additional Reading:
Textbook 5-6 pg. 273
Real World Application
Physics
The discovery of the lever allowed force
ancient people to accomplish great
tasks like building Stonehenge and
the Egyptian pyramids. A lever
multiplies the force applied to an
object. One example of a lever is
the nutcracker. Use the SAS or SSS
Inequality to explain how to operate
the nutcracker.
fulcrum
load
force
Challenge Problem
Suppose that plane F bisects AC at B for a
point D in F, DC > DA. What can you
conclude about the relationship between AC
and plane F?
F
D
C
A
B
Challenge Homework
Pgs. 277-279
#21, 23, 27,
29, 39
Parallelograms
TEKS/TAKS:
1.a, 2.b, 4.a, 5.a, 7.a, 7.b, 9.b
Objective:
You will recognize and apply the properties of a
parallelogram, and you will find the probability of an event.
Used in:
Transportation and interior design
Vocabulary:
Quadrilaterals, parallelogram, diagonals, probability
Additional Reading:
Textbook 6-1 pg. 291
Real World Application
Language
In a commercial for CompuServe Computer
Discount House, the announcer says that he
thought a parallelogram was a “telegram for
gymnasts at the Olympics.” Look up the
suffix “gram” in a dictionary. Then make a
conjecture about why a parallelogram is
named as it is.
Challenge Problem
Consider parallelogram RSTV. As the
measure of angle R decreases, what must
happen to the measure of angle V? What is
the maximum measure for angle V?
Explain your reasoning.
Challenge Homework
Pgs. 295-297
#25, 33, 41,
45, 47
Tests for Parallelograms
TEKS/TAKS:
1.a, 1.b, 2.b, 4.a, 5.a, 7.a, 7.b, 9.b, 10.a
Objective:
You will recognize and apply the conditions that ensure a
quadrilateral is a parallelogram and you will identify and use
subgoals in writing proofs.
Used in:
Engineering and arts
Vocabulary:
Identifying subgoals
Additional Reading:
Textbook 6-2 pg. 298
Real World Application
Drafting
Before computer drawing programs became
available, blueprints for buildings or mechanical
parts were drawn by hand. One of the tools
drafters used, a parallel ruler, is shown above.
Holding one of the bars in place and moving the
other allowed the drafter to draw a line parallel to
the first in many positions on the page. Why does
the parallel ruler guarantee that the second line will
be parallel to the first?
Challenge Problem
Ellen claims she has invented a new geometry theorem:
“A diagonal of a parallelogram bisects its angles”.
She gives the following proof:
Given: parallelogram MATH with diagonal MT
Prove: MT bisects <AMH and <ATH
Proof: Since MATH is a parallelogram, MH is congruent to AT and
MA is congruent to HT. Since MT is congruent to MT, triangle MHT is
congruent to MAT by SSS. Therefore, <1 is congruent to <2 and <3
is congruent to <4.
Do you think Ellen’s new theorem is true? Why or why not?
Challenge Homework
Pgs. 301-303
#17, 23, 25,
39, 43
Rectangles
TEKS/TAKS:
1.a, 1.b, 2.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 5.a, 9.b
Objective:
You will recognize and apply the properties of rectangles.
Used in:
Architecture and sports
Vocabulary:
Rectangle
Additional Reading:
Textbook 6-3 pg. 306
Real World Application
Music
Compact discs (CDs) are circular in shape
but packaged in rectangular cases. Why do
you think rectangular packaging is used?
Challenge Problem
Write a two-column proof.
Given: WXYZ
<1 and <2 are complementary
Prove: WXYZ is a rectangle.
X
W
Y
1
2
Z
Challenge Homework
Pgs. 309-312
#11, 13, 27,
37, 49
Squares and Rhombi
TEKS/TAKS:
1.a, 2.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 5.a, 7.b, 9.b
Objective:
You will recognize and apply the properties of squares and
rhombi.
Used in:
Art and construction
Vocabulary:
Rhombus, rhombi, square
Additional Reading:
Textbook 6-4 pg. 313
Real World Application
Construction
The opening for the reinforced doors of a
storage shed is shaped like a square.
Identify the quadrilaterals that make up the
doors. Explain why you think the doors are
shaped as they are.
Challenge Problem
Use a ruler to draw a quadrilateral with
perpendicular diagonals that is not a
rhombus.
Challenge Homework
Pgs. 317-319
#33, 45, 53,
55, 57
Kites
TEKS/TAKS:
2.a, 2.b, 9.b
Objective:
You will recognize and apply the properties of kites.
Used in:
Art and construction
Vocabulary:
Kite
Additional Reading:
Textbook 6-4B pg. 320
Real World Application
Construction
Draw AC in kite ABCD. Use a protractor to
measure the angles formed by the
intersection of AC and BD. Measure the
interior angles of kite ABCD. Are any
congruent?
Challenge Problem
Given kite ABCD, label the intersection of
AC and BD as point E. Find the lengths
of AE, BE, CE, and DE. How are they
related?
Challenge Homework
Pg. 320
Write at least five original
conjectures about kites.
Justify your conjectures.
Trapezoids
TEKS/TAKS:
1.a, 2.b, 3.c, 3.e, 4.a, 5.a, 7.a
Objective:
You will recognize and apply the properties of trapezoids.
Used in:
Sailing and engineering
Vocabulary:
Trapezoid, bases, legs, base angles, isosceles trapezoid
Additional Reading:
Textbook 6-5 pg. 321
Real World Application
Architecture
Tray ceilings make a room appear to be
larger than it actually is. The trays are make
using wood or drywall panels. What type of
quadrilaterals are used to make the tray
ceiling shown above?
Challenge Problem
Draw a trapezoid with two right angles.
Label the bases and the legs.
Try to draw an isosceles trapezoid with
two right angles. Is it possible? Explain
why or why not.
Challenge Homework
Pgs. 325-328
#27, 31, 37,
43, 47
Using Proportions
TEKS/TAKS:
1.a, 2.b, 3.e, 4.a, 5.a, 9.b, 11.b
Objective:
You will recognize and use ratios and proportions and you will
apply properties of proportions.
Used in:
Movie props, literature, and meteorology
Vocabulary:
Ratios, proportion, cross products, means, extremes
Additional Reading:
Textbook 7-1 pg. 338
Real World Application
Cartography
The scale on a map indicates that 1.5
centimeters represents 200 miles. If the
distance on the map between Norfolk,
Virginia, and Atlanta, Georgia, measures 2.4
centimeters, how many miles apart are the
cities?
Challenge Problem
At each gas stop on a recent trip, Keshia
recorded the distance she traveled since
the last stop and the amount of gas she
purchased for her midsize car. Find the
average number of miles per gallon.
Stops
Distance (mi)
Gas (gal)
1
351
13
2
275
11
3
362.5
12.5
4
372
12
Challenge Homework
Pgs. 342-344
#25, 29, 33,
39, 47
Exploring Similar Polygons
TEKS/TAKS:
1.a, 1.b, 2.b, 3.e, 4.a, 5.a, 7.b, 9.b, 11.b
Objective:
You will identify similar figures and solve problems involving
similar figures.
Used in:
Cartography, gardening, and construction work
Vocabulary:
Similar figures, dilation
Additional Reading:
Textbook 7-2 pg. 346
Real World Application
Construction
On the floor plan for a new house, one inch
represents 18 feet. If the living room is 3/4
inch by 5/8 inch, what are the dimensions?
Challenge Problem
Use what you know about slope and
distance to show that two triangles are
similar. Triangle ABC has vertices A(0,0),
B(12,0), and C(6,9). Triangle DEF has
vertices D(18,0), E(26,0), and F(22,6).
Challenge Homework
Pgs. 351-353
#29, 31, 33,
37, 43
Identifying Similar Triangles
TEKS/TAKS:
1.a, 1.b, 2.b, 3.e, 4.a, 5.a, 9.b, 11.b
Objective:
You will identify similar triangles and use similar triangles to
solve problems.
Used in:
Surveying, forestry
Vocabulary:
None
Additional Reading:
Textbook 7-3 pg. 354
Real World Application
Surveying (pg. 360)
Mr. Cardona uses a carpenter’s square, an
instrument used to draw right angles, to find the
distance across a stream. He puts the square on
top of a pole that is high enough to sight along OL
to point P across the river. Then he sights along
ON to point M. If MK is 2.5 feet and OK = 5.5 feet,
find the distance KP across the stream.
Challenge Problem
Is it possible that triangle ABC is not
similar to triangle RST and that triangle
RST is not similar to triangle EFG, but
that triangle ABC is similar to triangle
EFG? Explain.
Challenge Homework
Pgs. 358-360
#17, 23, 25,
27, 39
Parallel Lines and
Proportional Parts
TEKS/TAKS:
1.a, 2.a, 2.b, 3.e, 4.a, 5.a, 7.a, 11.b, 11.c
Objective:
You will use proportional parts of triangles to solve problems
and to divide a segment into congruent parts.
Used in:
History and navigation
Vocabulary:
None
Additional Reading:
Textbook 7-4 pg. 362
Real World Application
Construction
Hai was building a large open
stairway and used wood strips
as decoration along the inside
wall. If the strips were spaced
along the bottom as shown in
the diagram above, at what
distance should he attach the
top of the strips if the strips are
to be parallel?
x
8 ft
y
z
2.5 ft
2 ft
1.5 ft
Challenge Problem
Draw any quadrilateral ABCD and
connect the midpoints E, F, G, H of the
sides in order. Determine what kind of
figure EFGH will be. Prove your claim.
Challenge Homework
Pgs. 367-369
#25, 33, 37,
39, 41
Parts of Similar Triangles
TEKS/TAKS:
1.a, 2.b, 3.b, 3.c, 3.e, 4.a, 5.a
Objective:
You will recognize and use the proportional relationships of
corresponding perimeters, altitudes, angle bisectors, and
medians of similar triangles.
Used in:
Photography, design, and art
Vocabulary:
None
Additional Reading:
Textbook 7-5 pg. 370
Real World Application
Design
Julian had a picture 18 centimeters by 24
centimeters that he wanted enlarged by 30%
and then have the inside of the frame edges
with navy blue piping. The store only had
110 centimeters of navy blue piping in stock.
Will this be enough piping to fit on the inside
edge of the frame? Explain.
Challenge Problem
Consider two rectangular
prisms shown at the right.
What ratios are necessary to
determine whether they are
similar? If the first prism has
dimensions that are three
times as large as the second,
will the volume also have a
ratio of 1 to 3? Explain why
or why not.
s
h
w
a
b
c
Challenge Homework
Pgs. 374-377
#21, 27, 31,
45, 47
Fractals and Self-Similarity
TEKS/TAKS:
1.a, 2.b, 4.a, 5.b, 11.a, 11.b, 11.d
Objective:
You will recognize and describe characteristics of fractals and
will solve problems by solving a simpler problem.
Used in:
Nature and art
Vocabulary:
Sierpinski triangle, fractal, self-similar, strictly self-similar,
solve a simpler problem
Additional Reading:
Textbook 7-6 pg. 378
Real World Application
Solve a Simpler Problem
Look at the diagonals in Pascal’s triangle. Find the sum of
the first 25 numbers in the outside diagonal. Find the sum
of the first 50 numbers in the second diagonal.
Challenge Problem
The fractal in the pictures at
the right is the space-filling
Hilbert curve.
Define the iterative process
used to generate the curve.
Why do you think it is called
“space filling”?
Challenge Homework
Pgs. 381-383
#7, 11, 17, 23,
27
Geometric Mean and the
Pythagorean Theorem
TEKS/TAKS:
1.a, 1.b, 2.b, 3.e, 4.a, 5.c, 8.c, 11.c
Objective:
You will find the geometric mean between two numbers, solve
problems involving relationships between parts of a triangle
and the altitude to its hypotenuse, and use the Pythagorean
Theorem and its converse.
Used in:
Architecture
Vocabulary:
Geometric mean, Pythagorean triple
Additional Reading:
Textbook 8-1 pg. 397
Real World Application
Motion Pictures
In the movie The Wizard of Oz, the Scarecrow is
looking for a brain. When the Wizard presents
him with a Doctor of Thinkology degree, the
Scarecrow immediately announces “The sum of
the square roots of any two sides of an isosceles
triangle is equal to the square root of the
remaining side.” Do you agree with the
“Scarecrow Theorem”? Explain.
Challenge Problem
Draw an acute triangle and an obtuse triangle.
In the acute triangle, draw an altitude to the longest
side.
In the obtuse triangle, draw an altitude from the
obtuse angle.
In either case, are the triangles formed by the
altitude similar to the original? Explain.
Challenge Homework
Pgs. 402-403
#27, 29, 31,
33, 43
Special Right Triangles
TEKS/TAKS:
1.a, 2.b, 4.a, 5.c, 11.c
Objective:
You will use the properties of 45-45-90 and 30-60-90
triangles.
Used in:
Sports, city planning, and landscaping
Vocabulary:
None
Additional Reading:
Textbook 8-2 pg. 405
Real World Application
City Planning
Granmichele is a city with a
population of about 15,000 people in
northern Sicily. The city has a
hexagonal design as shown by the
aerial view at the right. Consider all
sides of the hexagons to be
congruent. Suppose the
perpendicular distance from the
center of the city to each side of the
one hexagon is 0.5 mile. Find the
perimeter of that hexagon.
Challenge Problem
Two parallel lines are cut by a transversal.
The transversal makes a 120° angle with one
of the parallel lines. A line bisects the 120°
angle. Another line bisects the consecutive
interior angle.
Draw the figure and show that the triangle
formed by the two angle bisectors and the
transversal is a 30-60-90 triangle.
Challenge Homework
Pgs. 409-411
#15, 21, 29,
33, 34
Ratios in Right Triangles
TEKS/TAKS:
1.a, 2.b, 4.a, 5.c, 9.b, 11.c
Objective:
You will find trigonometric ratios using right triangles and you
will solve problems using trigonometric ratios.
Used in:
Aviation, medicine, and astronomy
Vocabulary:
Trigonometry, trigonometric ratio, sine, cosine, tangent
Additional Reading:
Textbook 8-3 pg. 412
Real World Application
Medicine
9.8 cm
Skin
A patient is being treated with
radiotherapy for a tumor that is
Organ
behind a vital organ. In order to
prevent damage to the organ, the
radiologist must angle the rays to the
tumor. If the tumor is 6.3 cm below Tumor
the skin and the rays enter the body
9.8 cm to the right of the tumor, find
the angle the rays should enter the
body to hit the tumor.
Challenge Problem
Draw a circle in Q1 of a coordinate plane with a radius of 1
unit. Each division on the x- and y-axes is 0.2 unit. Right
triangles are formed by drawing a vertical line from the point
on the x-axis to the circle, the connecting that point with the
origin. Use the Pythagorean Theorem and trigonometry to
complete the table of values for each triangle.
Length of
hypotenuse
X-value
1
0.2
1
0.4
1
0.6
1
0.8
Y-value
Sin O
Cos O
Challenge Homework
Pgs. 416-418
#35, 39, 47,
49, 53
Angles of Elevation and Depression
TEKS/TAKS:
1.a, 2.b, 4.a, 11.c
Objective:
You will use trigonometry to solve problems involving angles
of elevation or depression.
Used in:
Aerospace, architecture, and meteorology
Vocabulary:
Angle of depression, angle of elevation
Additional Reading:
Textbook 8-4 pg. 420
Real World Application
Literature
“In the Adventures of Sherlock Holmes: The Adventures of the Musgrave
Ritual”, Sherlock Holmes uses trigonometry to solve the mystery. To find
a treasure, he must determine where the end of the shadow of an elm tree
was located at a certain time of day. Unfortunately, the elm had been cut
down, but Mr. Musgrave remembers that his tutor required him to
calculate the height of the tree as part of his trigonometry class. Mr.
Musgrave tells Sherlock Holmes that the tree was exactly 64 feet.
Sherlock needs to find the length of the shadow at a time of day when the
shadow from an oak tree is a certain length. The angle of elevation of the
sun at this time of day is 33.7°. What was the length of the shadow of the
elm?
Challenge Problem
Imagine that a fly and an ant are in
one corner of a rectangular box. The
end of the box is 4 inches by 6
inches, and the diagonal across the
bottom of the box makes an angle of
21.8° with the longer edge of the box.
There is food in the corner opposite
the insects.
What is the shortest distance the fly
must fly to get to the food?
What is the shortest distance the ant
must crawl to get to the food?
21.8°
Food
Insects
4 in
6 in
Challenge Homework
Pgs. 423-425
#21, 27, 31,
33, 35
Exploring Circles
TEKS/TAKS:
1.a, 1.b, 2.b, 4.a, 7.a, 9.b, 9.c
Objective:
You will identify and use parts of circles and solve problems
involving the circumference of a circle.
Used in:
Surveying, sports, and space travel
Vocabulary:
Circle, center, radius, chord, diameter, circumference, pi
Additional Reading:
Textbook 9-1 pg. 446
Real World Application
Culture
About one thousand years ago, wooden poles
formed a gigantic circle 125 meters across in
southern Illinois. The structure was a giant
solar calendar that kept track of the seasons
and the movement of the sun for Native North
Americans. What was the circumference of
this structure?
Challenge Problem
Use the Triangle Inequality Theorem to show diameter SA is
the longest chord in circle P. That is, write a paragraph proof
that shows SA > KR. (Hint: Draw PK and PR).
S
P
A
R
K
Challenge Homework
Pgs. 450-451
#31, 33, 37,
43, 47
Angles and Arcs
TEKS/TAKS:
1.a, 2.b, 4.a, 5.a, 8.b, 9.b, 9.c
Objective:
You will recognize major arcs, minor arcs, semicircles, and central
angles, you will find measures of arcs and central angles, and you
will solve problems by making circle graphs.
Used in:
Data display, clocks, statistics
Vocabulary:
Central angle, arc, minor arc, major arc, semicircle, adjacent arcs,
arc length, concentric circles, similar circles, congruent circles,
congruent arcs
Additional Reading:
Textbook 9-2 pg. 452
Real World Application
Clocks
The Floral Clock in Frankfort,
Kentucky, has a diameter of 34 feet.
The hands on the clock form a
central angle with the circular
timepiece. Suppose the hour hand
is on the 10 (A) and the minute
hand is on the 2 (C) and the vertex
is B.
Find the measure of the central
angle ABC.
Find the arc length of the minor arc.
Challenge Problem
Draw a diagram to explain how it is possible
for two central angles to be congruent, yet
their corresponding minor arcs are NOT
congruent.
Challenge Homework
Pgs. 456-457
#41, 43, 49,
53, 55
Arcs and Chords
TEKS/TAKS:
1.a, 2.a, 2.b, 4.a, 5.a, 9.b, 9.c
Objective:
You will recognize and use relationships among arcs, chords,
and diameters.
Used in:
Geology
Vocabulary:
Arc of the chord, inscribed polygon
Additional Reading:
Textbook 9-3 pg. 459
Real World Application
Road Signs
Jodi wants to create a new yield
sign that inscribes the yellow
isosceles triangle in a circle. Draw
an isosceles triangle and explain
how Jodi could use the theorems
from this lesson to find the center of
the circle that contains the yellow
yield sign.
Challenge Problem
Draw a circle R and choose a point P on the circle.
Now draw six chords with P as one endpoint. Label
the other endpoints A, B, C, D, E and F. Make a
conjecture as to how the lengths of each chord are
related to their distances from the center of the
circle. Explain your reasoning.
Challenge Homework
Pgs. 463-465
#33, 37, 39,
43, 55
Inscribed Angles
TEKS/TAKS:
1.a, 2.a, 2.b, 4.a, 5.a, 9.c
Objective:
You will recognize and find measures of inscribed angles and
apply properties of inscribed figures.
Used in:
Civil engineering and carpentry
Vocabulary:
Inscribed angle, intercepted arc
Additional Reading:
Textbook 9-4 pg. 466
Real World Application
Civil Engineering
A
A civil engineer uses the formula
l = (2*pi*r*m)/360 to calculate
the length l of a curve for a road,
given a radius r and a central
angle measure m. Find the
length of the road from A to B in
the diagram at the right. Round
to the nearest foot.
l
220 ft
70 ft
220 ft
B
Challenge Problem
Can an isosceles trapezoid be inscribed in a
circle? Write a brief paragraph explaining
your reasoning.
Challenge Homework
Pgs. 470-473
#27, 35, 45,
65, 67
Tangents
TEKS/TAKS:
5.a, 9.b, 9.c
Objective:
You will recognize tangents and use properties of tangents.
Used in:
Astronomy and aerospace
Vocabulary:
Tangent, point of tangency, interior, exterior, common internal
tangent, common external tangent, tangent segments
Additional Reading:
Textbook 9-5 pg. 475
Real World Application
Olympics
At the 1996 Summer
Olympics, Anthony
Washington of Aurora,
Colorado, finished fourth in
the discus event with a
throw of 65.42 meters. If he
wound up in a circular
pattern to throw the discus,
along a tangent line (path)
to the circle, use the figure
to find the radius of the
circle.
x
x
67.13 m
65.42 m
Challenge Problem
A unit circle is a circle with a radius of 1. In
the figure, circle O is a unit circle with QR
tangent to circle O at R. Use the right triangle
trigonometry definition of tangent to find tan
Q
theta.
1
1 R
Challenge Homework
Pgs. 480-482
#33, 35, 39,
43, 55
Tangents
TEKS/TAKS:
5.a, 9.b, 9.c
Objective:
You will recognize tangents and use properties of tangents.
Used in:
Astronomy and aerospace
Vocabulary:
Tangent, point of tangency, interior, exterior, common internal
tangent, common external tangent, tangent segments
Additional Reading:
Textbook 9-5 pg. 475