# How Do Dilations Map Lines, Rays, and Circles?

### New York State Common Core Math Geometry, Module 2, Lesson 8

Student Outcomes

• Students prove that a dilation maps a ray to a ray, a line to a line and a circle to a circle.

How Do Dilations Map Lines, Rays, and Circles?

Classwork

Opening Exercise

a. Is a dilated ray still a ray? If the ray is transformed under a dilation, explain how.
b. Dilate the 𝑃𝑄 by a scale factor of 2 from center O.
i. Is the figure 𝑃′𝑄′ a ray?
ii. How, if at all, has the ray 𝑃𝑄 been transformed?
iii. Will a ray always be mapped to a ray? Explain how you know

Example 1

Will a dilation about center 𝑂 and scale factor 𝑟 = 1 map 𝑃𝑄 to 𝑃′𝑄′? Explain.

Example 2

The line that contains 𝑃𝑄 does not contain point 𝑂. Does a dilation about center 𝑂 and scale factor 𝑟 ≠ 1 map every point of 𝑃𝑄 onto a point of 𝑃′𝑄′?

Example 3

The line that contains 𝑃𝑄 contains point 𝑂. Does a dilation about center 𝑂 and scale factor 𝑟 map 𝑃𝑄 to 𝑃′𝑄′?
a. Examine the case where the endpoint 𝑃 of 𝑃𝑄 coincides with the center 𝑂 of the dilation.
b. Examine the case where the endpoint 𝑃 of 𝑃𝑄 is between 𝑂 and 𝑄 on the line containing 𝑂, 𝑃, and 𝑄.
c. Examine the remaining case where the center 𝑂 of the dilation and point 𝑄 are on the same side of 𝑃 on the line containing 𝑂, 𝑃, and 𝑄.

Lesson Summary

• DILATION THEOREM FOR RAYS: A dilation maps a ray to a ray sending the endpoint to the endpoint.
• DILATION THEOREM FOR LINES: A dilation maps a line to a line. If the center 𝑂 of the dilation lies on the line or if the scale factor 𝑟 of the dilation is equal to 1, then the dilation maps the line to the same line. Otherwise, the dilation maps the line to a parallel line.
• DILATION THEOREM FOR CIRCLES: A dilation maps a circle to a circle and maps the center to the center.

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