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?
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.
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:
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:
Examples 2–3 Construct the line of reflection across which each image below was reflected
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.
Example 5 Now try a slightly more complex figure. Reflect 𝐴𝐵𝐶𝐷 across 𝐸𝐹̅̅̅̅.
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