options: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Verify experimentally the properties of rotations, reflections, and translations:
- Lines are taken to lines, and line segments to line segments of the same length.
- Angles are taken to angles of the same measure.
- Parallel lines are taken to parallel lines.
problem_activity: Warm-up
The purpose of this warm-up is to ensure students understand that
they can infer the coordinates of a point based on knowing the coordinates of points on the same horizontal and vertical lines;
the length of a horizontal or vertical line segment can be determined based on the coordinates of its endpoints. Each of these is important background knowledge for this lesson.
Launch
Give students 2 minutes of quiet think time. Ask them to share their reasoning with a partner followed by a whole-class discussion.
Student Facing
[IMG]
Find each of the following and explain your reasoning:
The length of segment
\(BE\)
.
The coordinates of
\(E\)
.
Student Response
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Anticipated Misconceptions
In order to find the length of segment
\(BE\)
, it is first necessary to find the lengths of
\(CD\)
,
\(AC\)
, and
\(AB\)
, and reason about side lengths in similar triangles. If students have trouble getting started, scaffold the work by suggesting they first find any
other
segment lengths in the diagram that they can find.
Activity Synthesis
There are three important things students should understand or recall as a result of working on the warm-up:
Points on the same vertical line have the same
\(x\)
-coordinate and points on the same horizontal line have the same
\(y\)
-coordinate.
In order to find the length of a vertical segment, you can subtract the
\(y\)
-coordinates of its endpoints.
The side lengths of triangle
\(ACD\)
are the same as the side lengths of the similar triangle
\(ABE\)
multiplied by the scale factor
\(AC \div AB\)
.
Display the image from the task. For each question, invite a student to share their reasoning. The displayed image should be used as a tool for gesturing. For example, if you start at the origin and want to navigate to point
\(D\)
, you would move 4 units to the right. You move the same distance to the right to navigate from the origin to
\(C\)
. Therefore,
\(C\)
must have the same
\(x\)
-coordinate as
\(D\)
.


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