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9-6 Dilations
You identified dilations and verified them as similarity transformations.
• Draw dilations.
• Draw dilations in the coordinate plane.
DefinitionsA 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.
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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.
Draw a DilationCopy 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.
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.
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.
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.
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)
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