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Understanding Dilations in Geometry: Scale Factor, Enlargement, Reduction

Learn about dilations in geometry, determining scale factors for transformations, effects on dimensions, and how to identify enlargement, reduction, or congruence. Practice examples included.

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Understanding Dilations in Geometry: Scale Factor, Enlargement, Reduction

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  1. Dilations Dilations Dilations Dilations Dilations Dilations Dilations Dilations Dilations Section 9.5

  2. Determine whether a dilation is an enlargement, a reduction, or a congruence transformation. • Determine the scale factor for a given dilation. • dilation • similarity transformation Standard 11.0 Students determine how changes in dimensions affect the perimeter, area,and volume of common geometric figures and solids. Lesson 5 MI/Vocab

  3. Dilation • Center: C • Scale Factor: r or k (the book uses both) • Def: A transformation that maps every point P to a point P’ so that the following properties are true: • P’ lies on • The scale factor is:

  4. Scale Factor • If r > 1, the dilation is called an enlargement. • If r < 1, the dilation is called a reduction. • If r = 1, the dilation is called a congruence transformation. • (Just like in chapter 7) A scale factor can also be found by comparing corresponding lengths in a ratio.

  5. Example #1 Determine the scale factor, then state if the dilation is an enlargement, reduction or congruence transformation. r < 1 reduction

  6. Example #2 Determine the scale factor, then state if the dilation is an enlargement, reduction or congruence transformation. r > 1 enlargement

  7. Theorem 9.2: If a dilation with center C and a scale factor of r transforms A to A’ and B to B’, then: A’B’ = |r|(AB) Algebraic hint: AB = Lesson 5 KC1

  8. Example #3 Given PQ = 1, Q’R’ = 4, use the scale factor to find P’Q’ and QR. 1

  9. Example #3 Given PQ = 1, Q’R’ = 4, use the scale factor to find P’Q’ and QR. 4

  10. A. Find the measure of the dilation image C′D′ using the given scale factor for CD = 15, r = 3. Determine Measures Under Dilations Answer: 45 Lesson 5 Ex1

  11. B. Find the measure of the preimage using the given scale factor if Multiply each side by Determine Measures Under Dilations Answer: 10.5 Lesson 5 Ex1

  12. A. Find the measure of the dilation image A′B′ using the given scale factor for AB = 16, r = –2. • A • B • C • D A. 2 B. 4 C. 8 D. 32 Lesson 5 CYP1

  13. B. Find the measure of the preimage using the given scale factor for A′B′ = 24, r = . • A • B • C • D A. 16 B. 36 C. 48 D. 18 Lesson 5 CYP1

  14. Which diagram shows the dilation image of ΔLMN with center C and ? • A • B • C • D A. B. C.D. Lesson 5 CYP2

  15. Theorem 9.3 If is the preimage of a dilation centered at the origin with a scale factor k, then the image is Multiply the coordinates by the scale factor. Lesson 5 TH2

  16. Dilations in the Coordinate Plane COORDINATE GEOMETRY Trapezoid EFGH has vertices E(–8, 4), F(–4, 8), G(8, 4) and H(–4, –8). Find the image of trapezoid EFGH after a dilation centered at the origin with a scale factor of Sketch the preimage and the image. Lesson 5 Ex3

  17. Dilations in the Coordinate Plane Answer: E'(–2, 1), F'(–1, 2), G'(2, 1), H'(–1, –2) Hint: If you did it right… Lesson 5 Ex3

  18. COORDINATE GEOMETRY Triangle ABChas 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 A. B.C. D. none of the above Lesson 5 CYP3

  19. Scale Drawing Sharetta built a frame for a photograph that is 20 centimeters by 25 centimeters. The frame measures 40 centimeters by 50 centimeters. Which scale factor did she use? Find the scale factor. frame lengthphoto length Simplify. Answer: Sharetta used a scale factor of 2 to build the frame. Lesson 5 Ex5

  20. Ruben is making a scale drawing of the front of his house. His house is 48 feet wide and 30 feet high at its highest point. Ruben decides on a dilation reduction factor of What size poster board will he need to make a complete drawing? • A • B • C • D A. 19 in. by 26 in. B. 22 in. by 30 in. C. 20.5 in. by 28 in. D. 16 in. by 29 in. Lesson 5 CYP5

  21. Homework: Pg 529: #1, 2, 6 – 8, 10 – 19 all, 26 – 32 even.

  22. Identify Scale Factor A. Determine the scale factor used for the dilation with center C. Determine whether the dilation is an enlargement, reduction, or congruence transformation. 6 2 Lesson 5 Ex4

  23. A.2; enlargement B. C. D.1; congruence transformation A. For the green preimage, determine the scale factor used for each dilation with center C. Determine whether the dilation is an enlargement, reduction, or congruence transformation. • A • B • C • D Lesson 5 CYP4

  24. A.2; enlargement B. C. D.3; enlargement B. For the green preimage,determine the scale factor used for each dilation with center C. Determine whether the dilation is an enlargement, reduction, or congruence transformation. • A • B • C • D Lesson 5 CYP4

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