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Example

Example. A quadrilateral ABCD has vertices A = (-7,4) B = (0, 3) C = (5, 1), and D = (-2, 2). It is translated by the vector . Graph ABCD and its translation image A’B’C’D’ on the coordinate plane. Dilations. Definitions.

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  1. Example • A quadrilateral ABCD has verticesA = (-7,4) B = (0, 3) C = (5, 1), and D = (-2, 2). • It is translated by the vector . Graph ABCD and its translation image A’B’C’D’ on the coordinate plane.

  2. Dilations

  3. Definitions • Scale Factor: The scale factor is the number we multiply the sides or coordinates of a figure by to get the new coordinates or sides. • Enlargement: When you multiply by a number whose absolute value is greater then 1. (Gets Bigger) • Reduction: When you multiply by a number whose absolute value is less than 1. (Gets Smaller)

  4. Identifying Dilations If P is not the center point C, then the image point P lies on . The scale factor k is a positive number such that k = , and k 1. CP CP CP Previously, you studied rigid transformations, in which the image and preimage of a figure are congruent. In this lesson, you will study a type of nonrigid transformation called a dilation, in which the image and preimage of a figure are similar. A dilation with center C and scale factor k is a transformation that maps every point P in the plane to a point P so that the following properties are true. If P is the center point C, then P=P.

  5. Identifying Dilations Reduction: k = = = Enlargement: k = = CP CP CP CP 5 2 1 2 3 6 Because PQR ~ P´Q´R´, is equal to the scale factor of the dilation. P´Q´ PQ The dilation is a reduction if 0 < k < 1 and it is an enlargement if k > 1. P P´ 6 5 P´ P 2 3 Q Q´ R • • C C R´ Q R´ Q´ R

  6. Identifying Dilations P 3 P´ 2 • C 2 3 2 3 Because = , the scale factor is k = . CP CP Identify the dilation and find its scale factor. SOLUTION This is a reduction.

  7. Identifying Dilations P´ 2 P P 3 P´ 1 2 • • C C 2 1 Because = , the scale factor is k = 2. CP CP Because = , the scale factor is k = . CP CP 2 3 2 3 Identify the dilation and find its scale factor. SOLUTION SOLUTION This is an enlargement. This is a reduction.

  8. Dilation in a Coordinate Plane Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4). Use the origin as the center and use a scale factor of . How does the perimeter of the preimage compare to the perimeter of the image? 1 2 C´ x 1 O 1 y A´ In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky). SOLUTION Because the center of the dilation is the origin, you can find the image of each vertex by multiplying its coordinates by the scale factor. D C A B D´ A(2, 2)  A´(1, 1) B(6, 2)  B´(3, 1) B´ • C(6, 4)  C´(3, 2) D(2, 4)  D´(1, 2)

  9. Dilation in a Coordinate Plane Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4). Use the origin as the center and use a scale factor of . How does the perimeter of the preimage compare to the perimeter of the image? 1 2 C´ O y 1 1 x A´ In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky). SOLUTION From the graph, you can see that the preimage has a perimeter of 12 and the image has a perimeter of 6. D C A B D´ A preimage and its image after a dilation are similar figures. B´ • Therefore, the ratio of the perimeters of a preimage and its image is equal to the scale factor of the dilation.

  10. Using Dilations in Real Life Shadow puppets have been used in many countries for hundreds of years. A flat figure is held between a light and a screen. The audience on the other side of the screen sees the puppet’s shadow. The shadow is a dilation, or enlargement, of the shadow puppet. When looking at a cross sectional view, LCP ~ LSH. = 59 74 12 SH LC LS CP SH = The shadow puppet shown is 12 inches tall (CP in the diagram). Find the height of the shadow, SH, for each distance from the screen. In each case, by what percent is the shadow larger than the puppet? LC = LP = 59 in.; LS = LH = 74 in. SOLUTION 59(SH) = 888 SH 15 inches

  11. Using Dilations in Real Life Shadow puppets have been used in many countries for hundreds of years. A flat figure is held between a light and a screen. The audience on the other side of the screen sees the puppet’s shadow. The shadow is a dilation, or enlargement, of the shadow puppet. When looking at a cross sectional view, LCP ~ LSH. SH CP 15 12 scale factor = = 1.25 So, the shadow is 25% larger than the puppet. LC = LP = 59 in.; LS = LH = 74 in.; SH15 inches SOLUTION To find the percent of size increase, use the scale factor of the dilation.

  12. Using Dilations in Real Life Shadow puppets have been used in many countries for hundreds of years. A flat figure is held between a light and a screen. The audience on the other side of the screen sees the puppet’s shadow. The shadow is a dilation, or enlargement, of the shadow puppet. When looking at a cross sectional view, LCP ~ LSH. = 66 74 12 SH LC LS CP SH = LC = LP = 66 in.; LS = LH = 74 in. SOLUTION 66 (SH) = 888 SH 13.45 inches

  13. Using Dilations in Real Life Shadow puppets have been used in many countries for hundreds of years. A flat figure is held between a light and a screen. The audience on the other side of the screen sees the puppet’s shadow. The shadow is a dilation, or enlargement, of the shadow puppet. When looking at a cross sectional view, LCP ~ LSH. SH CP 13.45 12 scale factor = = 1.12 So, the shadow is 12% larger than the puppet. LC = LP = 66 in.; LS = LH = 74 in.; SH13.45 inches SOLUTION To find the percent of size increase, use the scale factor of the dilation.

  14. Example of a dilation • The dilation is a reduction if 0<k<1 and it is an enlargement if k > 1 5 2 1 2

  15. Example 1 • Identify the dilation and find its scale factor C

  16. Example 1 • Identify the dilation and find its scale factor C Reduction, Because The scale factor is k = ⅔ Enlargement, Because The scale factor is k = 2

  17. Example 2Dilation in a Coordinate Plane • Draw a dilation of a rectangle ABCD with vertices A(1,1), B(3,1), C(3,2) and D(1,2). Use the origin as the center and use a scale factor of 2. How does the perimeter of the preimage compare to the perimeter of the image?

  18. Example 2Dilation in a Coordinate Plane • Draw a dilation of a rectangle ABCD with vertices A(1,1), B(3,1), C(3,2) and D(1,2). Use the origin as the center and use a scale factor of 2. How does the perimeter of the preimage compare to the perimeter of the image? • Preimage has a perimeter of 6 • Image has a perimeter of 12 • The perimeter was enlarged by a factor of 2

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