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Eli wants to show that circle J is similar to circle K using transformations. What combination of transformations would map circle J onto circle K to show that the circles are similar? Select two answers that, when combined, show that the circles are similar. Dilate with a scale factor of . Dilate with a scale factor of 3. Dilate with a scale factor of 2. Translate 6 units right and 6 units down. Translate 6 units left and 6 units up. Rotate 90° clockwise.

Eli wants to show that circle J is similar to circle K using transformations. What combination of transformations would map circle J onto circle K to show that the circles are similar? Select two answers that, when combined, show that the circles are similar. Dilate with a scale factor of . Dilate with a scale factor of 3. Dilate with a scale factor of 2. Translate 6 units right and 6 units down. Translate 6 units left and 6 units up. Rotate 90° clockwise.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Understanding Transformations to Show Similarity of Circles** Eli wants to demonstrate that circle \( J \) is similar to circle \( K \) through the use of transformations. To achieve this, the correct combination of transformations must be identified to map circle \( J \) onto circle \( K \), thus proving their similarity. To explore this concept, consider the following options. Select **two answers** that, when combined, will show that the circles are indeed similar: 1. ☐ Dilate with a scale factor of \(\frac{1}{2}\). 2. ☐ Dilate with a scale factor of 3. 3. ☐ Dilate with a scale factor of 2. 4. ☐ Translate 6 units right and 6 units down. 5. ☐ Translate 6 units left and 6 units up. 6. ☐ Rotate 90° clockwise. **Explanation:** - **Dilation** involves resizing a shape by expanding or contracting it from a particular point using a specific scale factor. - **Translation** involves shifting the shape in a straight path in any direction without rotating or resizing it. - **Rotation** involves turning the shape around a fixed point. When working with circles, similarity can be shown by using dilation (which must be in the correct proportion) and proper translation to relocate the circle without altering its shape. Thus, identifying the ideal transformations from the given options is key to demonstrating their similarity.
Transcribed Image Text:**Understanding Transformations to Show Similarity of Circles** Eli wants to demonstrate that circle \( J \) is similar to circle \( K \) through the use of transformations. To achieve this, the correct combination of transformations must be identified to map circle \( J \) onto circle \( K \), thus proving their similarity. To explore this concept, consider the following options. Select **two answers** that, when combined, will show that the circles are indeed similar: 1. ☐ Dilate with a scale factor of \(\frac{1}{2}\). 2. ☐ Dilate with a scale factor of 3. 3. ☐ Dilate with a scale factor of 2. 4. ☐ Translate 6 units right and 6 units down. 5. ☐ Translate 6 units left and 6 units up. 6. ☐ Rotate 90° clockwise. **Explanation:** - **Dilation** involves resizing a shape by expanding or contracting it from a particular point using a specific scale factor. - **Translation** involves shifting the shape in a straight path in any direction without rotating or resizing it. - **Rotation** involves turning the shape around a fixed point. When working with circles, similarity can be shown by using dilation (which must be in the correct proportion) and proper translation to relocate the circle without altering its shape. Thus, identifying the ideal transformations from the given options is key to demonstrating their similarity.
### Study the Diagram of Circles K and J --- #### Circle K - **Center:** \((-3, 3)\) - **Radius:** 6 #### Circle J - **Center:** \((3, -3)\) - **Radius:** 3 --- ### Diagram Explanation The provided diagram illustrates two circles, labeled \(K\) and \(J\), plotted on a Cartesian coordinate system with a grid overlaid. 1. **Circle \(K\):** - The center of circle \(K\) is at the point \((-3, 3)\). This is marked with a blue dot. - The radius of circle \(K\) is 6 units. This circle is drawn to reflect the specified radius, extending 6 units from its center in all directions. 2. **Circle \(J\):** - The center of circle \(J\) is at the point \((3, -3)\). This is marked with a red dot. - The radius of circle \(J\) is 3 units. This circle is drawn to reflect the specified radius, extending 3 units from its center in all directions. ### Features of the Graph: - **Coordinate Axes:** The x-axis (horizontal) and y-axis (vertical) intersect at the origin \((0, 0)\). - **Grid:** The grid is square and divided in units for precise plotting of the circles. - **Arrows:** Arrows at the ends of both axes indicate the positive and negative directions along the axes. This graph is an illustrative tool for understanding the relationships and positions of the circles \(K\) and \(J\) in a coordinate plane. Each circle's size and placement help highlight concepts in geometry and the coordinate system.
Transcribed Image Text:### Study the Diagram of Circles K and J --- #### Circle K - **Center:** \((-3, 3)\) - **Radius:** 6 #### Circle J - **Center:** \((3, -3)\) - **Radius:** 3 --- ### Diagram Explanation The provided diagram illustrates two circles, labeled \(K\) and \(J\), plotted on a Cartesian coordinate system with a grid overlaid. 1. **Circle \(K\):** - The center of circle \(K\) is at the point \((-3, 3)\). This is marked with a blue dot. - The radius of circle \(K\) is 6 units. This circle is drawn to reflect the specified radius, extending 6 units from its center in all directions. 2. **Circle \(J\):** - The center of circle \(J\) is at the point \((3, -3)\). This is marked with a red dot. - The radius of circle \(J\) is 3 units. This circle is drawn to reflect the specified radius, extending 3 units from its center in all directions. ### Features of the Graph: - **Coordinate Axes:** The x-axis (horizontal) and y-axis (vertical) intersect at the origin \((0, 0)\). - **Grid:** The grid is square and divided in units for precise plotting of the circles. - **Arrows:** Arrows at the ends of both axes indicate the positive and negative directions along the axes. This graph is an illustrative tool for understanding the relationships and positions of the circles \(K\) and \(J\) in a coordinate plane. Each circle's size and placement help highlight concepts in geometry and the coordinate system.
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