9-6 Dilations You identified dilations and verified them as similarity transformations. Draw dilations. Draw dilations in the coordinate plane.

Definitions A dilation or scaling is a similarity transformation that enlarges or reduces a figure proportionally with respect to a center point and a scale factor. Scale factor is the amount of enlargement or reduction needed to get one figure from the other when two figures are similar. This is the number you multiply the side lengths of one figure by to create a second similar figure. p. 674

Scale Factor If the scale factor is greater than 1, the similar figure is an enlargement. If the scale factor is less than 1, it is a reduction.

Scale Factors

Draw a Dilation Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3. Since k > 1, the dilation is an enlargement of trapezoid PQRS. Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3. Locate P', Q', R', and S' so that Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3.

How to Draw a Dilation 1. Connect the center point and each vertex of the pre-image. 2. Measure (in centimeters) each segment you just drew. 3. Multiply each segment by the scale factor. 4. Measure that length to find your new point from the center point. 5. Connect the dots and label the image with prime notation.

Which diagram shows the dilation image of ΔLMN with center C and ? A. B. C. D.

PUPPETS To create the illusion of a “life-sized” image, puppeteers sometimes use a light source to show an enlarged image of a puppet projected on a screen or wall. Suppose that the distance between a light source L and the puppet is 24 inches (LP). To what distance PP' should you place the puppet from the screen to create a 49.5-inch tall shadow (I'M') from a 9-inch puppet? Understand This problem involves a dilation. The center of the dilation is L, LP = 24 in., IM = 9 in., I'M' = 49.5 in. You are asked to find PP'. Plan Find the scale factor of the dilation from the preimage IM to the image I'M'. Use the scale factor to find LP and then use LP and LP' to find PP'.

Solve The scale factor k of the enlargement is the ratio of the length on the image to a corresponding length on the preimage. Scale factor of image image = I'M', preimage = IM Divide. Use this scale factor of 5.5 to find LP'. LP' = k(LP) Definition of dilation = 5.5(24) k = 5.5 and LP = 24 = 132 Multiply. Use LP' and LP to find PP'. LP + PP' = LP' Segment Addition

Use LP' and LP to find PP'. LP + PP' = LP' Segment Addition 24 + PP' = 132 LP = 24 and LP' = 132 PP' = 108 Subtract 24 from each side. Answer: So, the puppet should be placed so that the distance from it to the screen (PP') is 108 inches. Check Since the dilation is an enlargement, the scale factor should be greater than 1. Since 5.5 > 1, the scale factor is reasonable.

Use this rule to find the image of a figure after a dilation centered at the origin (0,0). p. 676

Trapezoid EFGH has vertices E(–8, 4), F(–4, 8), G(8, 4) and H(–4, –8) Trapezoid EFGH has vertices E(–8, 4), F(–4, 8), G(8, 4) and H(–4, –8). Graph the image of EFGH after a dilation centered at the origin with a scale factor of ¼ Multiply the x- and y-coordinates of each vertex by the scale factor, ¼ Graph the preimage and image. Answer: E'(–2, 1), F'(–1, 2), G'(2, 1), H'(–1, –2)

How to Draw a Dilation 1. Multiply each ordered pair by the scale factor. 4. Locate the new ordered pair 5. Connect the dots and label the image with prime notation.

Triangle ABC has vertices A(–1, 1), B(2, –2), and C(–1, –2) Triangle ABC has vertices A(–1, 1), B(2, –2), and C(–1, –2). Find the image of ΔABC after a dilation centered at the origin with a scale factor of 2. Sketch the preimage and the image. A. B. C. D. none of the above

9-6 Assignment Page 677, 10, 12, 16, 18, 21-23