Unit 1, Lesson 14

Defining Rotations

Geometry

14.1

Warm-up

Which pairs of angles appear congruent? How could you check?

<p>6 angles. Angles A B C and R S L open to the right, angles D E F, G H L, M Z Y, and P B K open up.</p>

14.2

Activity

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because . . . .”
Continue to ask questions until you have enough information to solve the problem.
When you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.

If your teacher gives you the data card:

Silently read the information on your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.

<p>Several points: B, C, D, F, G, H, I, J, K, L, M, N, U, V, W.</p>

14.3

Activity

Draw a segment. Label the endpoints and .
1. Rotate segment clockwise around center by 90 degrees. Label the new endpoint .
2. Connect to , and lightly shade in the resulting triangle.
3. What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.
Draw a segment. Label the endpoints and .
1. Rotate segment counterclockwise around center by 30 degrees. Label the new endpoint .
2. Rotate segment counterclockwise around center by 30 degrees. Label the new endpoint .
3. Connect to , and lightly shade in the resulting triangle.
4. What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.

Student Lesson Summary

A rotation is a transformation with a center, an angle, and a direction (clockwise or counterclockwise).

Here is how a rotation with a center point , an angle that measures , and a counterclockwise direction transforms a point :

The rotation sends point to a point on the circle with a radius of length .
The angle measures and is counterclockwise around the circle from .

<p>Two images of angle t degrees. First shows center C, and t degrees angle, congruent line segments PC and P’C and a counterclockwise rotation taking P to P’. Second shows angle t degrees.</p>

If the direction were clockwise instead, then would be clockwise around the circle of radius . If and are in the same place, then the rotation sends to on the circle with a radius of 0 units, so points , , and are all in the same place.

A rotation is a rigid transformation that is defined by a center, an angle, and a direction. It takes one point on a circle to another point, using a given center. The two radii—the one from the center to the original point and the one from the center to the image—make the angle of rotation.

In this figure, is the image of after a counterclockwise rotation of using the point as the center.

In this figure, quadrilateral is rotated counterclockwise using the point as the center.

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Defining Rotations

Warm-up

Activity

Activity

Student Lesson Summary