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Lesson 1 Centered! Develop Understanding

Jump Start

In previous tasks, we have defined a circle as the set of all points in a plane equidistant from a point called its center. We have also defined concentric circles as two or more circles that have the same center. In this unit, several other terms related to circles will be useful. The following activity will help you become familiar with these words.

Working with a partner, do the following:

  • Read a definition. Draw an example of the term on one of the two circles and label it with the new vocabulary word. Both partners should decide together if the proposed drawing matches the definition.

  • Continue this process until all words have been illustrated and labeled on the diagram.

Note: You might want to use colored pencils to help annotate your diagrams.

  • Chord—a line segment whose endpoints lie on a circle.

  • Secant—a line that intersects a circle at exactly two points.

  • Tangent—a line that intersects a circle at exactly one point.

  • Diameter—a chord that passes through the center of a circle.

  • Radius—a line segment with one endpoint at the center of a circle and the other endpoint on the circle.

Note: the words radius and diameter are also used to refer to the lengths of these segments.

  • Arc—a portion of a circle.

  • Central angle—an angle whose vertex is at the center of a circle and whose sides pass through a pair of points on the circle.

  • Inscribed angle—an angle formed when two chords, two secant lines, or a secant and tangent line, intersect at a point on a circle.

  • Intercepted arc—the portion of a circle that lies between two lines, rays, or line segments that intersect the circle.

A circle
A circle

Learning Focus

Find the center of rotation for a given pair of pre-image/image figures.

How can you locate the center of a given rotation using only the figures given as the pre-image and final image?

To use a compass to draw a circle, you start with the center point and radius, and then construct the circle. But what if the circle has already been drawn? How do you determine the location of its center and its radius?

Open Up the Math: Launch, Explore, Discuss

Franklin and Fabio know how to construct the image of a rotation when given the center and angle of rotation, but today they have encountered a different issue: How do you find the center of rotation when a rotated image and its pre-image are given? They decide to explore this idea with their friends, Miranda and Mateo.

Each pair of friends creates a “puzzle” for the other pair by sketching a drawing on graph paper in which a rotation of a figure is shown, but the center of rotation is not marked. The other pair has to figure out where the center of rotation is located. Here are the “puzzles” they created for each other.

Point A(4,3) Point A'(0,5)x–10–10–10–5–5–5555101010y–10–10–10–5–5–5555101010000

Franklin and Fabio’s Puzzle

Triangle ABC A(6,5) B(5,0) C(9,-2) Triangle A'B'C' A'(-2,5) B'3,4), C'(5,8)x–10–10–10–5–5–5555101010y–10–10–10–5–5–5555101010000

Miranda and Mateo’s Puzzle

Miranda and Mateo think that the puzzle they have been given is too easy since it only consists of a single rotated point and its pre-image.

Miranda says, “The center of rotation is at the midpoint , halfway between the image and pre-image points, and the point has been rotated .”

Mateo disagrees, “The center of rotation is at the point since both the image and pre-image points are away from origin. I’ll need to use a protractor to find the angle of rotation.”

Laughing, Fabio says, “You’re both wrong. We didn’t use either or as the center of rotation when we created the puzzle.”

Miranda replies, “I can see how both of our points can be the center of rotation, but now I think that with a single image/pre-image pair of points, any point can be the center of rotation.”

1.

This puzzle has turned out to be more challenging than Miranda and Mateo thought. List at least three additional points that could be considered the center of rotation, and justify your choices.

2.

What do you think about Miranda’s last statement, “Any point can be the center of rotation?” Do you agree or disagree? If you agree, explain. If you disagree, what would be a better statement to make about the set of points that can be used as the center of rotation for a single rotated point and its pre-image?

3.

Now examine the puzzle Miranda and Mateo gave to Franklin and Fabio. Find the center of rotation for this puzzle; or, if you believe there can be more than one center of rotation, describe how all of the possible centers of rotation are related.

4.

Describe, illustrate, and prove your process works for finding the center of rotation of a figure using the perpendicular bisectors of segments connecting several pre-image/image pairs of points. That is, if you make any claims in your description for finding the center of rotation, make sure you provide a proof of your claims. Use correct mathematical vocabulary. Here are words associated with circles that you defined in the Jump Start. Some of these terms may be useful in your written description of how to find the center of rotation.

(Note: Not all of these words will be useful for answering problem 4, but they will be useful in future tasks, so they are given here for reference.)

  • Circle

  • Concentric circles

  • Chord

  • Secant

  • Tangent

  • Diameter

  • Radius

  • Arc

  • Central angle

  • Inscribed angle

  • Intercepted arc

My process for finding the center of rotation of a figure consisting of several image/pre-image pairs of points, along with my justification for why the process works:

Ready for More?

Prove the following theorem: In a circle, the perpendicular bisector of a chord bisects the central angle formed by the radii drawn to the endpoints of the chord.

Takeaways

Given two points in a plane, one point can be carried onto the other by a rotation centered at

Given a figure consisting of several pre-image/image pairs of points, you can find the center of rotation by:

Describe:

Illustrate:

Based on these ideas, we can prove that a central angle of a circle can be bisected by:

Describe:

Illustrate:

Vocabulary

Lesson Summary

In this lesson, we examined the question: How do you find the center of a rotation if you are only given the pre-image and image figures? This question led us to a strategy involving perpendicular bisectors of segments, as we drew upon the theorem that points on the perpendicular bisector of a segment are equidistant from the endpoints, and its converse, points equidistant from the endpoints of a segment lie on the perpendicular bisector. We were able to find a connection between perpendicular bisectors and the rotation transformation. This connection allowed us to prove a theorem about chords and the central angles formed by drawing radii to the endpoints of the chords.

Retrieval

1.

Trapezoids and are similar. Find the scale factor and the center of dilation.

Trapezoid WXZY W(1.5,2.5) X(1,1) Y(4,1) Z(3.5,2.5); Trapezoid W'X'Z'Y' W'(3.8,3.5), X'(3.6,3), Y'(4.6,3), Z'(4.5,3.5)111222333444555666111222333444000

2.

Find the area and the circumference of a circle with a radius of . Leave in your answer.