Reflections

New York State Common Core Math Geometry, Module 1, Lesson 14

Worksheets for Geometry

Student Outcomes

• Students learn the precise definition of a reflection.
• Students construct the line of reflection of a figure and its reflected image.
• Students construct the image of a figure when provided the line of reflection.

Reflections

Classwork

Exploratory Challenge

Think back to Lesson 12 where you were asked to describe to your partner how to reflect a figure across a line. The greatest challenge in providing the description was using the precise vocabulary necessary for accurate results. Let’s explore the language that yields the results we are looking for.

△ 𝐴𝐵𝐶 is reflected across 𝐷𝐸 and maps onto △ 𝐴′𝐵′𝐶′.

Use your compass and straightedge to construct the perpendicular bisector of each of the segments connecting 𝐴 to 𝐴′, 𝐵 to 𝐵′, and 𝐶 to 𝐶′. What do you notice about these perpendicular bisectors?

Label the point at which 𝐴𝐴′ intersects 𝐷𝐸 as point 𝑂. What is true about 𝐴𝑂 and 𝐴′𝑂? How do you know this is true?

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Discussion

You just demonstrated that the line of reflection between a figure and its reflected image is also the perpendicular bisector of the segments connecting corresponding points on the figures. In the Exploratory Challenge, you were given the pre-image, the image, and the line of reflection. For your next challenge, try finding the line of reflection provided a pre-image and image.

Example 1

Construct the segment that represents the line of reflection for quadrilateral 𝐴𝐵𝐶𝐷 and its image 𝐴′𝐵′𝐶′𝐷′.

What is true about each point on 𝐴𝐵𝐶𝐷 and its corresponding point on 𝐴′𝐵′𝐶′𝐷′ with respect to the line of reflection?

Notice one very important fact about reflections. Every point in the original figure is carried to a corresponding point on the image by the same rule—a reflection across a specific line. This brings us to a critical definition:

REFLECTION: For a line 𝑙 in the plane, a reflection across 𝑙 is the transformation 𝑟𝑙of the plane defined as follows:

1. For any point 𝑃 on the line 𝑙, 𝑟𝑙 (𝑃) = 𝑃, and
2. For any point 𝑃 not on 𝑙, 𝑟𝑙 (𝑃) is the point 𝑄 so that 𝑙 is the perpendicular bisector of the segment 𝑃𝑄.

If the line is specified using two points, as in 𝐴𝐵 , then the reflection is often denoted by 𝑟𝐴𝐵. Just as we did in the last lesson, let’s examine this definition more closely:

• A transformation of the plane—the entire plane is transformed; what was once on one side of the line of reflection is now on the opposite side;
• 𝑟𝑙(𝑃) = 𝑃 means that the points on line 𝑙 are left fixed—the only part of the entire plane that is left fixed is the line of reflection itself;
• 𝑟𝑙(𝑃) is the point 𝑄—the transformation 𝑟𝑙 maps the point 𝑃 to the point 𝑄;
• The line of reflection 𝑙 is the perpendicular bisector of the segment 𝑃𝑄—to find 𝑄, first construct the perpendicular line 𝑚 to the line 𝑙 that passes through the point 𝑃. Label the intersection of 𝑙 and 𝑚 as 𝑁. Then locate the point 𝑄 on 𝑚 on the other side of 𝑙 such that 𝑃𝑁 = 𝑁𝑄.

Examples 2–3 Construct the line of reflection across which each image below was reflected

2. You have shown that a line of reflection is the perpendicular bisector of segments connecting corresponding points on a figure and its reflected image. You have also constructed a line of reflection between a figure and its reflected image. Now we need to explore methods for constructing the reflected image itself. The first few steps are provided for you in this next stage.

Example 4
The task at hand is to construct the reflection of △ 𝐴𝐵𝐶 over ̅𝐷𝐸̅̅̅. Follow the steps below to get started; then complete the construction on your own.

1. Construct circle 𝐴: center𝐴, with radius such that the circle crosses ̅𝐷𝐸̅̅̅ at two points (labeled 𝐹 and 𝐺).
2. Construct circle 𝐹: center 𝐹, radius 𝐹𝐴 and circle 𝐺: center 𝐺, radius 𝐺𝐴. Label the (unlabeled) point of intersection between circles 𝐹 and 𝐺 as point 𝐴′. This is the reflection of vertex 𝐴 across ̅𝐷𝐸̅̅̅.
3. Repeat steps 1 and 2 for vertices 𝐵 and 𝐶 to locate 𝐵′ and 𝐶′.
4. Connect 𝐴′, 𝐵′, and 𝐶′ to construct the reflected triangle.
   Things to consider:
   When you found the line of reflection earlier, you did this by constructing perpendicular bisectors of segments joining two corresponding vertices. How does the reflection you constructed above relate to your earlier efforts at finding the line of reflection itself? Why did the construction above work?

Example 5 Now try a slightly more complex figure. Reflect 𝐴𝐵𝐶𝐷 across 𝐸𝐹̅̅̅̅.

Lesson Summary

• A reflection carries segments onto segments of equal length.
• A reflection carries angles onto angles of equal measure.

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