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Common Core For Geometry

Worksheets for Geometry, Module 1, Lesson 14

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. Letβs explore the language that yields the results we are looking for.

β³ π΄π΅πΆ is reflected across π·πΈ and maps onto β³ π΄β²π΅β²πΆβ².

Use your compass and straightedge to construct the perpendicular bisector of each of the segments connecting π΄ to π΄β², π΅ to π΅β², and πΆ to πΆβ². What do you notice about these perpendicular bisectors?

Label the point at which π΄π΄β² intersects π·πΈ as point π. What is true about π΄π and π΄β²π? How do you know this is true?

**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 π΄π΅πΆπ· and its image π΄β²π΅β²πΆβ²π·β².

What is true about each point on π΄π΅πΆπ· and its corresponding point on π΄β²π΅β²πΆβ²π·β² 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 ruleβa reflection across a specific line. This brings us to a critical definition:

**REFLECTION**: For a line π in the plane, a reflection across π is the transformation ππof the plane defined as follows:

- For any point π on the line π, ππ (π) = π, and
- For any point π not on π, ππ (π) is the point π so that π is the perpendicular bisector of the segment ππ.

If the line is specified using two points, as in π΄π΅ , then the reflection is often denoted by ππ΄π΅. Just as we did in the last lesson, letβs examine this definition more closely:

- A transformation of the planeβthe entire plane is transformed; what was once on one side of the line of reflection is now on the opposite side;
- ππ(π) = π means that the points on line π are left fixedβthe only part of the entire plane that is left fixed is the line of reflection itself;
- ππ(π) is the point πβthe transformation ππ maps the point π to the point π;
- The line of reflection π is the perpendicular bisector of the segment ππβto find π, first construct the perpendicular line π to the line π that passes through the point π. Label the intersection of π and π as π. Then locate the point π on π on the other side of π such that ππ = ππ.

**Examples 2β3**
Construct the line of reflection across which each image below was reflected

- 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 β³ π΄π΅πΆ over Μ
π·πΈΜ
Μ
Μ
. Follow the steps below to get started; then complete
the construction on your own.

- Construct circle π΄: centerπ΄, with radius such that the circle crosses Μ π·πΈΜ Μ Μ at two points (labeled πΉ and πΊ).
- Construct circle πΉ: center πΉ, radius πΉπ΄ and circle πΊ: center πΊ, radius πΊπ΄. Label the (unlabeled) point of intersection between circles πΉ and πΊ as point π΄β². This is the reflection of vertex π΄ across Μ π·πΈΜ Μ Μ .
- Repeat steps 1 and 2 for vertices π΅ and πΆ to locate π΅β² and πΆβ².
- Connect π΄β², π΅β², and πΆβ² 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 π΄π΅πΆπ· across πΈπΉΜ
Μ
Μ
Μ
.

**Lesson Summary**

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

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