standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
problem_activity: Activity
Students have seen that the lengths of corresponding segments in a figure and its scaled copy vary by the same scale factor. Here, they learn that in such a pair of figures,
any
corresponding distances—not limited to lengths of sides or segments—are related by the same scale factor. The side lengths of the polygons in this task cannot be easily determined, so students must look to other distances to compare.
Students must take care when they identify corresponding vertices and distances. As students work, urge them to attend to the order in which points or segments are listed.
If students are not sure what to make out of the values in the table (for the second question), encourage them to consider the corresponding distances of two figures at a time. For example, ask: What do you notice about the corresponding vertical distances in
\(IJKL\)
and
\(EFGH\)
? What about the corresponding horizontal distances in those two figures?
Teacher Notes for IM 6–8 Math Accelerated
Adjust this activity to 15 minutes. Conclude the activity synthesis by displaying this image:
[IMG]
Ask, “Here are two quadrilaterals. Kiran says that Polygon
\(EFGH\)
is a scaled copy of
\(ABCD\)
, but Lin disagrees. Do you agree with either of them?” (Lin is correct, because the corresponding distances are not multiplied by the same number. Kiran may have noticed that the corresponding angles are equal and thought this meant the polygons are scaled copies. I noticed that the scale factors for the corresponding sides are not the same.
\(AB\)
and
\(EF\)
are related by a scale factor of
\(\frac23\)
, but
\(DC\)
and
\(HG\)
are related by a scale factor of
\(\frac12\)
.)
After a brief quiet think time, select students to share their responses. The goal of this discussion is to make clear that angle measurements and distances are both important when deciding whether two polygons are scaled copies.
Launch
Arrange students in groups of 2. Ask if they can tell the lengths of segments
\(GF\)
or
\(DC\)
from the grid (without using rulers). Explain that they will explore another way to compare length measurements in scaled copies.
Give students 2–3 minutes of quiet work time for the first two questions, and 1 minute to discuss their responses with a partner before continuing on to the last question.
Student Facing
Each of these polygons is a scaled copy of the others. You already checked their corresponding angles.
[IMG]
The side lengths of the polygons are hard to tell from the grid, but there are other
corresponding distances
that are easier to compare. Identify the distances in the other two polygons that correspond to
\(DB\)
and
\(AC\)
, and record them in the table.
| quadrilateral | distance that                       corresponds to                       \(DB\) | distance that                       corresponds to                       \(AC\) |
|---------------|---------------------------------------------------------------------------------|---------------------------------------------------------------------------------|
|   \(ABCD\)    |                                                                                 |                                                                                 |
|   \(EFGH\)    |                                                                                 |                                                                                 |
|   \(IJKL\)    |                                                                                 |                                                                                 |
Look at the values in the table. What do you notice?
Pause here so your teacher can review your work.
The larger figure is a scaled copy of the smaller figure.
[IMG]
If
\(AE = 4\)
, how long is the corresponding distance in the second figure? Explain or show your reasoning.
If
\(IK = 5\)
, how long is the corresponding distance in the first figure? Explain or show your reasoning.
Student Response
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Anticipated Misconceptions
Students may list the corresponding vertices for distances in the wrong order. For example, instead of writing
\(IK\)
as the distance corresponding to
\(DB\)
, they may write
\(KI\)
. Remind students of the corresponding points by asking, “Which vertex in
\(IJKL\)
corresponds to
\(D\)
? Which corresponds to
\(B\)
?” and have them match the order of the vertices accordingly.
Activity Synthesis
Display the completed tables for all to see. To highlight how all distances in a scaled copy (not just the side lengths of the figure) are related by the same scale factor, discuss:
How does the vertical distance in
\(ABCD\)
compare to that in
\(EFGH\)
? How do the horizontal distances in the two polygons compare? Do the pairs of vertical and horizontal distances share the same scale factor?
How do the vertical distances in
\(EFGH\)
and
\(IJKL\)
compare? What about the horizontal distances? Is there a common scale factor? What is that scale factor?
What scale factor relates the corresponding lengths and distances in the two drawings of the letter W?
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between figures. For example, mark corresponding line segments using the same color.
Supports accessibility for: Visual-spatial processing
Speaking: MLR7 Compare and Connect.
Use this routine to call attention to the different ways students may identify scale factors. Display the following statements: “The scale factor from
\(EFGH\)
to
\(IJKL\)
is 3,” and “The scale factor from
\(EFGH\)
to
\(IJKL\)
is
\(\frac13\)
.” Give students 2 minutes of quiet think time to read and consider whether either or both of the statements are correct. Invite students to share their initial thinking with a partner before selecting 2–3 students to share with the class. In this discussion, listen for and amplify any comments that refer to the order of the original figure and its scaled copy, as well as those who identify corresponding vertices and distances. Draw students’ attention to the different ways to describe the relationships between scaled copies and the original figure.
Design Principle(s): Maximize meta-awareness


aligns: no