Key Concept Overview: Grade 8 - Module 3 - Topic B - Lessons 8-12

GRADE 8 | MODULE 3 | TOPIC B | LESSONS 8–12
KEY CONCEPT OVERVIEW
In Topic B, students formally define similarity and investigate the properties of similar objects. Looking specifically
at triangles, students learn to tell whether two triangles are similar using techniques other than dilation or the basic
rigid motions. After determining that two triangles are similar, students use equivalent ratios to find the unknown
side lengths of a triangle. Finally, students apply their knowledge of similarity to real-world tasks, such as finding
heights of buildings and determining distances that are too large to measure with a typical measuring tool.
You can expect to see homework that asks your child to do the following:
▪▪ Describe a dilation followed by a sequence of rigid motions that would map one shape onto another.
▪▪ Determine whether two objects are similar using angle-angle criterion or proportional side relationships.
▪▪ Use dilation and rigid motion to prove that similarity is symmetric and transitive.
▪▪ Given that two triangles are similar, solve for the unknown side length.
▪▪ Use similar triangle relationships to solve problems with real-world contexts.
SAMPLE PROBLEM (From Lesson 12)
A geologist wants to determine the distance across the widest part of a nearby lake. The geologist marked off specific
points around the lake so that the line containing segment DE would be parallel to the line containing segment BC.
Segment BC is selected specifically because it is the widest part of the lake. Segment DE is selected specifically
because it is a short enough distance to measure easily. The geologist sketched the situation, as shown below.
a. Has the geologist done enough so far to use similar triangles to
help measure the widest part of the lake? Explain your answer.
Yes, based on the sketch, the geologist found a center of
dilation at point A. The geologist marked points around the
lake that, when connected, make parallel lines. The triangles
are similar by the angle-angle (AA) criterion. Corresponding
angles of parallel lines are equal in measure, and the
measure of ∠DAE is equal to itself. Since there are two pairs
of corresponding angles that are equal, V DAE ~V BAC .
b. The geologist made the following measurements: |DE| = 5 feet, |AE| = 7 feet, and |EC| = 15 feet. Does she
have enough information to complete the task? If so, determine the length across the widest part of the lake.
If not, state what additional information is needed.
Yes, there is enough information about the similar triangles to determine the distance across the
widest part of the lake using the AA criterion.
x 22
Let x represent the length of segment BC; then, 5 = 7 .
We are looking for the value of x that makes the fractions equivalent. Therefore, 7x = 110 and x ≈ 15.7.
The distance across the widest part of the lake is approximately 15.7 feet.
Additional sample problems with detailed answer steps are found in the Eureka Math Homework Helpers books. Learn more at GreatMinds.org.
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GRADE 8 | MODULE 3 | TOPIC B | LESSONS 8–12
HOW YOU CAN HELP AT HOME
You can help at home in many ways. Here are some tips to help you get started.
▪▪ Similar triangles are often used to solve problems in the real world. Help your child master the information
needed to determine whether two triangles are similar. Review facts about angle relationships from Grade
8 Module 2 Topic C, where students explored the angles created when a line (called a transversal) passes
through parallel lines. Students should be familiar with corresponding, alternate interior, alternate exterior,
and supplementary angles from Module 2.
▪▪ Play vocabulary games. Work with your child to create vocabulary and key concept cards using the Terms
sections of this and previous Parent Tip Sheets. Write a term or concept on the front of each card and the
definition on the back. Then take turns with your child drawing a card, defining the term or concept on
the front, and checking the answer on the back. You might compete to be the first to provide five correct
answers. You might also use the board from a favorite board game and advance around it by defining terms
and concepts correctly.
TERMS
Angle-angle criterion: Two triangles are similar if two angles of one triangle are congruent (equal in
measure) to two angles from the other triangle.
Proportional: Two quantities (such as lengths or widths of objects) are proportional when they have the
same relative size in relation to each other. Here is an example of the notation you could use to show that the
side lengths of two shapes are proportional. If triangle ABC is similar to triangle FDE, then we can write the
following:
AB FD
=
BC DE

Similar/Similarity: Two objects are similar if there is a dilation followed by a sequence of rigid motions that
would map one object onto another. Similar shapes preserve angle measures, and the lengths of their sides are
proportional. We use the symbol ~ to represent similarity.
Symmetric property: Similar in reasoning to the commutative properties of addition and multiplication, the
symmetric property says that if A ~ B (A is similar to B), then B ~ A (B is similar to A).
Transitive property of similarity: This property states that one similar figure can be substituted for another
similar figure. If A ~ B (A is similar to B) and B ~ C (B is similar to C), then A ~ C (A is similar to C).
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GRADE 8 | MODULE 3 | TOPIC B | LESSONS 8–12

KEY CONCEPT OVERVIEW

SAMPLE PROBLEM (From Lesson 12)

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HOW YOU CAN HELP AT HOME

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