Geometry
4.5 Warm Up Day 1 Use the graph to find the indicated length. 1. Find the length of 2. Find the length of DE and EF BC and AC
4.5 Warm Up Day 2 Plot the points in a coordinate plane. Then determine whether the quadrilaterals are congruent. 1. A(-3, 4), B(-3, 7), C(3, -4), D(3, -1) 2. E(7, -2), F(2, -2), G(3, -4), H(5, -4) 3. I(9, 2), J(0, 2), K(6, 9), L(6, 2) 4. M(7, -9), N(7, 0), P(8, -3), Q(-1, -3)
4.5 Essential Question What does it mean to dilate a figure?
Goals Identify Dilations Make drawings using dilations.
Rigid Transformations Rotations Translations These were isometries: The pre-image and the image were congruent.
Dilation Dilations are non-rigid transformations. The pre-image and image are similar, but not congruent.
Dilation Enlargement
Dilation Reduction
Dilation R S C Center of Dilation T
Dilation CR 2CR R CR S R C Center of Dilation T
Dilation CR 2CR CS R CR 2CS S R CS S C Center of Dilation T
Dilation CR 2CR CS R CR 2CS S R CS S C CT Center of Dilation T 2CT CT T
Dilation CR RST ~ R S T 2CR CS R CR 2CS S R CS S C CT Center of Dilation T 2CT CT T
Dilation Definition A dilation with center C and scale factor k is a transformation that maps every point P to a point P so that the following properties are true: 1. If P is not the center point C, then the image point P lies on CP. The scale factor k is an integer such that k 1 and CP' k = CP 2. If P is the center point C, then P = P. 3. The dilation is a reduction if 0 < k < 1, and an enlargement if k > 1.
Dilation CR Enlargement 2CR CS R CR 2CS S R CS S C CT Center of Dilation T 2CT CT CR' CS ' CT ' 2 Scale Factor CR CS CT 1 T
Example 1 What type of dilation is this? F Reduction G F G C K H K H
Example 1 What is the scale factor? F 45 G k k FG ' ' 15 FG 45 FK ' ' 12 FK 36 1 3 1 3 36 12 F 15 G C Notice: k < 1 Reduction K H K H
Example 2 Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement k = CP CP k = 12 8 k = 3 2 Notice: k > 1 Enlarge ment
Your Turn Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement k = CP CP k = 18 30 k = 3 5 Notice: k < 1 Reduction
Remember: The scale factor k is S. F. = Image Length Pre image Length If 0 < k < 1 it s a reduction. If k > 1 it s an enlargement.
Coordinate Geometry Use the origin (0, 0) as the center of dilation. The image of P(x, y) is P (kx, ky). Notation: P(x, y) P (kx, ky). Read: P maps to P prime You need graph paper, a ruler, pencil.
Example 3 Graph ABC with A(1, 1), B(3, 6), C(5, 4). B C A
Example 3 B Using a scale factor of k = 2, locate points A, B, and C. P(x, y) P (kx, ky). A(1, 1) A (2 1, 2 1) = A (2, 2) B(3, 6) B (2 3, 2 6) = B (6, 12) C(5, 4) C (2 5, 2 4) = C (10, 8) B C C A A
Example 3 B Draw A B C. C B C A A
Example 3 B You re done. Notice that rays drawn from the center of dilation (the origin) through every preimage point also passes through the image point. B C C A A
Example 4 Graph quad. KLMN with K( 4, 8), L(0, 8), M(4, 4), and N( 4, 4) and its image after a dilation with a scale factor of 3 4 x, y 3 4 x, 3 4 y K( 4, 8) L(0, 8) M(4, 4) N( 4, 4) K ( 3, 6) L (0, 6) M (3, 3) N ( 3, 3)
Your Turn T(0, 12) Draw RSTV with R(0, 0) S( 6, 3) T(0, 12) V(6, 3) S(-6, 3) V(6, 3) R(0, 0)
Your Turn T(0, 12) Draw R S T V using a scale factor of k = 1/3. T (0, 4) S(-6, 3) V(6, 3) S (-2, 1) V (2, 1) R(0, 0) R (0, 0)
Your Turn T(0, 12) R S T V is a reduction. T (0, 4) S(-6, 3) V(6, 3) S (-2, 1) V (2, 1) R(0, 0) R (0, 0)
Negative Scale Factor In the coordinate plane, you can have scale factors that are negative numbers. When this occurs, the figure is dilated and rotates 180. The following is still true If 0 < k < 1 it s a reduction. If k > 1 it s an enlargement.
Example 5 Graph FGH with vertices F( 4, 2), G( 2, 4), and H( 2, 2) and its image after a dilation with a scale factor of 1 2 x, y 1 2 x, 1 2 y F( 4, 2) G( 2, 4) H( 2, 2) F (2, 1) G (1, 2) H (1, 1)
Your Turn Graph PQR with vertices P(1, 2), Q(3, 1), and R(1, 3) and its image after a dilation with a scale factor of 2. x, y 2x, 2y P(1, 2) Q(3, 1) R(1, 3) P ( 2, 4) Q ( 6, 2) R ( 2, 6)
Example 6 You are making your own photo stickers. Your photo is 4 inches by 4 inches. The image on the stickers is 1.1 inches by 1.1 inches. What is the scale factor of this dilation? S. F. = Image Length Pre image Length S. F. = 1.1 4 S. F. = 11 40
Your Turn An optometrist dilates the pupils of a patient s eyes to get a better look at the back of the eyes. A pupil dilates from 4.5 millimeters to 8 millimeters. What is the scale factor of this dilation? S. F. = S. F. = 8 4.5 S. F. = 16 9 Image Length Pre image Length = 80 45
Example 7 You are using a magnifying glass that shows the image of an object that is six times the object s actual size. Determine the length of the image of the spider seen through the magnifying glass. S. F. = 6 = x 1.5 Image Length Pre image Length x = 1.5 (6) x = 9 cm
Your Turn You are using a magnifying glass that shows the image of an object that is six times the object s actual size. The image of a spider is shown at the left. Find the actual length of the spider. S. F. = Image Length Pre image Length 6 = 12.6 x 6 1 = 12.6 x 6x = 12.6 x = 2.1 cm
Summary A dilation creates similar figures. A dilation can be a reduction or an enlargement. If the scale factor is less than one, it s a reduction, and if the scale factor is greater than one it s an enlargement. A negative scale factor is the same as a dilation with a 180 rotation. The scale factor is found using S. F. = Image Pre image
Assignment