Download Investigation 1: Similarity

Lesson One: Reasoning About Similar Triangles
Think About the Situation:
The Statue of Liberty in New York’s Harbor has a nose that
is 4 feet 6 inches long (Measurements taken from How
They Built the Statue of Liberty by M. Shapiro, Random
House, 1985).
a. How could you use your own nose and arm
length to determine the length of one of her
arms?
b. Pick two other body measurements and find the
approximate length that these measurements
should be on the Statue of Liberty.
c. What are the limitations of your measurements?
d. Examine in what you did with the three examples
in question a and b, How as your work the same
in the three cases? How did it change from case
to case?
Investigation 1: Similarity
As you work on the problems of this investigation, look for answers to the following questions:
How can you test whether two polygons are similar?
How can you create a polygon similar to a given polygon?
Does the order that you record the scale factor matter, why?
1. Cameras record images digitally and on film. When a
camera lens is positioned perpendicular to the plane
containing an object, the object and its recorded image
are similar. This photograph was taken with a digital
camera.
a. Describe how you could use information in the
photograph to help determine the actual
dimensions of the face of the cell phone.
b. What are those dimensions?
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
2. Two polygons with the same number of sides are similar provided their corresponding angles
have the same measures and the ratios of lengths of the corresponding sides is a constant.
In the diagram above, quadrilateral A’B’C’D’~quadrilateral ABCD. The ~ symbol means “is
similar to.”
m∠A' = m∠A, m∠B' = m∠B, m∠C ' = m∠C, m∠D ' = m∠D
A' B' 35 5
=
=
AB 14 2
B'C ' 25 5
=
=
BC 10 2
C ' D ' 15 5
=
=
CD
6 2
D ' A' 10 5
=
=
DA
4 2
5
AB.
2
5
or equivalently B'C ' = BC.
2
5
or equivalently C' D ' = CD.
2
5
or equivalently D' A' = DA.
2
or equivalently A' B' =
5
is called the scale factor from quadrilateral ABCD to quadrilateral A’B’C’D’. It
2
scales (multiplies) the length of each side of quadrilateral ABCD to produce the length of the
corresponding side of quadrilateral A’B’C’D’.
a. What is the scale factor from quadrilateral A’B’C’D to quadrilateral ABCD?
b. If two pentagons are similar, describe how to find the scale factor from the smaller
The constant
pentagon to the larger pentagon. Then describe how to find the scale factor from the
larger pentagon to the smaller pentagon.
c. Suppose PQR ~XYZ and the scale factor from PQR to XYZ is
3
. Write as many
4
mathematical statements as you can about pairs of corresponding angles and about
pairs of corresponding sides. Compare your statements with other students. Resolve
any differences.
3. As a class, form a “truth line” about each of the following conjectures. At one end of
the line will be “always true” and on the opposite end will be “always false,” in the
middle represents “sometimes.” As your teacher reads the following statements, place
yourself on the segment. Be as precise as you can.
a. All isosceles right triangles are similar.
b. All equilateral triangles are similar.
c. All squares are similar.
d. All regular hexagons are similar.
e. All circles are similar.
f. All rhombi are similar.
g. All rectangles are similar.
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
4. For each of the following, assume that in each pair of figures is similar with the same
orientation. Find all missing measurements.
a.
b.
c.
5. The Dutch artist M.C.Escher created the
mathematically inspired print shown as
an interesting twist on using a basic
shape, in this case, a lizard, to cover a
portion of the plane. It also represents
an early attempt to depict infinity. The
diagram beside the lizards is a portion of
the framework Escher used in his design
and △ABC is an isosceles right
triangle. Assume BC = 2 units.
a. Compare the markings on the
sides and angles of the framework diagram and explain why these make
sense in relation to the lizard picture. Carefully explain why the marking are
correct.
b. Determine if each statement below is correct. If so, explain why and give the
scale factor from the first triangle to the second triangle. If the statement is
not correct, explain why.
i. △1 ~ △3
ii. △2 ~ △6
iii. △4 ~ △6
iv. △8 ~ △3
v. △9 ~ △1
Summarize the Mathematics:
In this investigation, you explored similarity of polygons with a focus on testing for similarity
for triangles.
a. Explain why not all rectangles are similar.
b. What is the fewest number of measurements needed to test if two rectangles are
similar? Explain.
c. Explain why any two regular n-gons are similar. How would you determine a scale
factor relating the two n-gons?
d. What needs to be verified before you can conclude that two triangles,
PQR and UVW , are similar?
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
Check Your Understanding:
Each triangle described in the table below is similar to △ABC. For each triangle, use this fact and the
additional information given to:
a. Identify the correspondences between its vertices and those of △ABC.
b. Determine the remaining table entries.
Shortest
Longest
Side Length Side Length
Triangle Angle Measure
m<A = 64
m<D = ?
m<G = ?
m<J = ?
o
o
m<B = 18
m<E = ?
m<H = ?
m<K= 18o
o
m<C = 98
m<F = 18o
m<I = ?
m<L= 98o
AC = 4.0
Third Side
Length
AB = 12.8
BC = 11.6
IG = 6.4
GH = 5.8
Scale
Factor from
△ABC
JL = 14.0
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
1
2
Investigation 1: Student Handout
4.
a.
5.
b.
c.
Check Your Understanding:
Shortest
Longest
Side Length Side Length
Triangle Angle Measure
o
m<A = 64
m<D =
m<G =
m<J =
o
m<B = 18
m<E =
m<H =
m<K= 18o
o
m<C = 98
m<F = 18o
m<I =
m<L= 98o
AC = 4.0
Third Side
Length
AB = 12.8
BC = 11.6
IG = 6.4
GH = 5.8
Scale
Factor from
△ABC
1
2
JL = 14.0
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
Investigation Two: Sufficient Conditions for Similarity of Triangles
In investigation one you worked with similar shapes and used their
similarity and scale factor to find “missing” measures. To ensure
similarity, you had to know quite a bit about conditions on the
measures of corresponding angles and sides. In this investigation, you
will explore minimal conditions (short-cuts) that will ensure that two
triangles are similar. As you work on the problems of this investigation,
look for answers to the following question:
What combinations of side or angle measures are sufficient to
determine that two triangles are similar?
Triangles are special polygons. Some of them have special names. A
triangle with at least two sides of equal length is called isosceles. If
all three sides have the same length, the triangle is called
equilateral. An equilateral triangle is a special kind of isosceles
triangle.
Triangles have some characteristics that not all polygons share.
1. Each pair of statements includes one version about triangles and another version about
polygons in general. For each pair, first decide whether the statement is true for
triangles. Then, try to find a counterexample that shows the statement is not true for
polygons in general.
a. Statements 1:
i. If two angles in one triangle equal two angles in another triangle, then
their third angles must be equal.
ii. If two angles in one polygon equal two angles in another polygon, then
their other angles must be equal.
b. Statements 2:
i. If two triangles have their corresponding angles equal, then the triangles
are similar.
ii. If two polygons have their corresponding angles equal, then the polygons
are similar.
c. Statements 3:
i. If two triangles have their corresponding sides proportional, then the
triangles are similar.
ii. If two polygons have their corresponding sides proportional, then the
polygons are similar.
d. Statements 4:
i. Every triangle with two equal sides also has two equal angles.
ii. Every polygon with two equal sides also has two equal angles.
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
From investigation one, you can conclude that two triangles are similar whenever you know
that two angles of one triangle are equal to two angles of the other triangle. Mathematicians
express this by saying that having two pairs of equal angles is sufficient for concluding that
two triangles are similar.
2. For the following given conditions use appropriate tools strategically to investigate if
the information given can be considered sufficient to conclude that the two triangles
are similar.
Start out as a skeptic – try to find two triangles that fit the condition but that are not
similar. In other words, look for a counterexample. If you find such triangles, you will
have shown that the particular combination is not sufficient to conclude that the
triangles are similar.
On the other hand, you may decide that a given condition is sufficient- that is, that
there are no counterexamples. In that case, try to explain why any two triangles that
fit the condition must be similar.
a. Condition 1: An angle of one triangle is equal to an angle of the other triangle.
b. Condition 2: A side of one triangle is proportional to a side of the other triangle.
c. Condition 3: A pair of sides of one triangle is proportional to a pair of sides of the
other triangle.
d. Condition 4: The three sides of one triangle are proportional to the three sides of
the other triangle.
e. Condition 5: An angle of one triangle is equal to an angle of the other triangle,
and a side of one triangle is proportional to a side of the other triangle.
f. Condition 6: A pair of sides of one triangle is proportional to a pair of sides of the
other triangle, and the angles between these pairs of sides are equal.
g. Condition 7: A pair of sides of one triangle is proportional to a pair of sides of the
other triangle, and an angle not between the pair in one triangle is equal to the
corresponding angle of the other triangle.
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
You have seen that triangles seem to be different from other polygons with regard to
similarity. Why are triangles special? The next few questions investigate this question.
3. Pick four lengths and form a quadrilateral using those lengths for the sides. Then try to
use the same four lengths to form a quadrilateral that is not similar to the first, is this
possibly? Explain.
4. Repeat question 3, starting with more than four lengths. That is, pick some lengths and
form a polygon using those lengths. Using the same lengths, try to form a polygon that
is not similar to the first. Is this possible, explain.
5. Start with three lengths and use them to form a triangle. As in questions 3 and 4, try to
use the same lengths to form a triangle that is not similar to the first. Is this possible,
explain.
6. Your work with triangles will often involve a
situation in which the similar triangles are
overlapping, or inside one another. Here is an
example of overlapping similar triangles. Do you
see the similar triangles? Redraw and re-label the
figure to show the separate triangles.
7. Using what you know about similar triangles, find the unknown lengths in these pairs
of overlapping similar triangles.
a.
b.
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
8. In course 2, you calculated the height of Chicago’s Bat Column and other structures
using trigonometry. Triangle similarity provides another method.
Suppose a mirror is placed on the ground as shown. You position yourself to see the
top of the sculpture reflected in the mirror. An important property of physics states that
in such a case, the angle of incidence is congruent to the angle of reflection.
a. Sketch and label a pair of triangles
which, if proven similar, could be
used to calculate the height of the
Bath Column. Write a similarity
statement relating the triangles you
identified.
b. Prove that the two triangle identified
are similar.
c. Add to your diagram the following measurements:
i. The ground distance between you and the mirror image of the top of the
column is 6 feet 5 inches.
ii. The ground distance between the mirror image and the base of the
column is 116 feet 8 inches.
iii. Assume that the distance from the ground up to your eyes is 66 inches.
iv. About how tall is the column?
d. How could you use a trigonometric ratio and the ground distance from the mirror
image to the column to calculate the height of the Bath Column? What
measurements would you need? How could you obtain it?
9. As part of their annual October outing to studying the changing colors of trees in
northern Main, several science club members from Poland Regional HS decided to test
what they were learning in math class by finding the width of the Penobscot River at a
particular point A as shown in the diagram. Pacing from point A, they located points D,
E and C as shown in the diagram as well.
a. How do you think they used
similarity to calculate the distance
AB? Be as precise as possible in your
answer.
b. What is your estimate of the width of
the river at point A?
c. Another group of students repeated
the measurement, again pacing from A to locate points D, E and C. In this case,
AD = 18m, DE = 26m, and AC = 20 m. Would you expect that this second group
got the same estimate for the width of the river as you did in part b? Explain.
d. What trigonometric ratio could be used to calculate the width of the river? What
measurements would you need and how could you obtain them?
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
10. Study the diagram below of △ABC. MN connects the midpoints of M and N of sides AB and
BC , respectively.
a. How does MN appear to be related to AC ?
b. How does the length of MN appear to be related to the length of AC ?
c. Use interactive geometry software or careful paper-and-pencil drawings to investigate if
your observations in parts a and b hold for triangles of different shapes. Compare your
findings with those of your classmates.
d. Complete the following statement:
If a line segment joins the midpoint of two sides of a triangle, then it......
e. Collaborating with others as needed, write a proof of the statement in part d. That
statement is often called the Midpoint Connector Theorem for Triangle.
Summarize the Mathematics:
In this investigation, you established three sets of conditions, each of which is sufficient to
prove that two triangles are similar.
a. State and illustrate with diagrams, the sets of conditions on side lengths
and/or angle measures of a pair of triangles that ensures the triangles are
similar.
b. What strategies are helpful in using similarity to solve problems in applied
context? To prove mathematical statements?
Check Your Understanding:
The design of homes and office buildings often requires roof trusses that are of different sizes but
have the same pitch. The trusses shown below are designed for roofs with solar hot-water collectors.
For part of a through e, suppose a pair of roof trusses have the given characteristics. Determine if
△ABC ~ △PQR. If so, explain how you know that the triangles are similar and given the scale factor
from △ABC to △PQR. If not, give a reason for your conclusion.
a. m<A=57o, m<B=38o, m<P=57o, m<R=95o
b. AB=12, BC=15, m<B=35o, PQ=16, QR=20,
m<Q=35o
c. AB=BC, m<B=m<Q, PQ=QR
d. AC=4, BC=16, BA=18, PR=10, QR=40,
QP=48
e. AB=12, m<A=63o, m<C=83o, m<P=63o,
m<Q=34o, PQ=12
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
Investigation 2: Student Handout
4. a.
b.
8.
9.
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
Teacher Notes:
CCSS Standards
Note: Most of this investigation comes from Core Plus, Course 3, Unit 3 Lesson 1
Investigation 1. You can reference the teacher guide for those answers until I get
back around to teacher notes.
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools
Adapted from Contemporary Mathematics in Context: Core Plus Course 3 and Interactive Mathematics
Program: Shadows by S Buckner for Buncombe County Schools