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Dilations

This guide covers the fundamental concepts of dilations in geometry, detailing definitions such as dilation, center, enlargement, reduction, and scale factor. It illustrates how dilations create similar figures, maintaining congruent angles and proportional sides. Engage with hands-on examples involving the dilation of triangle JKL, exploring different scale factors and their effects on coordinates. Additional exercises are provided to classify enlargements and reductions, reinforcing understanding through practical applications and calculations.

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Dilations

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  1. Dilations

  2. Vocabulary • Dilation: The image created by enlarging or reducing a figure. • Center: The center of a dilation is a fixed point used for measurement when altering the size of the figure. • Enlargement: An image that is larger than the original figure. An enlargement has a scale factor greater than 1. • Reduction: An image that is smaller than the original figure. A reduction has a scale factor between 0 and 1

  3. A dilation is ALWAYS similar to the original figure. So the corresponding angles are congruent and corresponding sides are proportional. • Scale Factor: The ratio of a length on the image to a length on the original figure is the scale factor of the dilation.

  4. Examples Graph JKL with vertices J (3,8), K (10,6), and L (8,2). Then graph its image J’K’L’ after a dilation with a scale factor of ½.

  5. Find the coordinates of the image of triangle JKL after a dilation with each scale factor. Then graph JKL and J’K’L’. • A) Scale Factor 3 • B) Scale Factor 1/3

  6. Find and Classify a Scale Factor • Quadrilateral V’Z’X’W’ is a dilation of quadrilateral VZXW. Find the scale factor of the dilation, and classify it as an enlargement or a reduction. Graph V’Z’X’W’ Graph VZXW V’ (-5, 5) V (-2,2) Z’ (-2.5, 7.5) Z (-1, 3) X’ (2.5, 6) X (1, 2.5) W’ (5, 2.5) W (2, 1)

  7. Write a ratio of the x-or the y-coordinate of one vertex of the DILATION to the x- or y-coordinate of the CORRESPONDING vertex of the original figure. • Let’s use V and V’ since we have whole numbers. • V (-2, 2) • V’ (-5, 5) Y-coordinate of V’ = 5 Y-coordinate of V 2

  8. Our scale factor is 5/2 • Based on our scale factor our image is an enlargement since 5/2 is greater than 1. • Remember 5/2 is an improper fraction so if we change this into a mixed number we get 2.5

  9. Carleta’s optometrist dilates her pupils by a factor of 5/3. if her pupil before dilation has a diameter of 5 milimeters, find the new diameter after her pupil is dilated

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