Pearson Unit 2 Topic 8: Transformational Geometry 8-7: Dilations Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (3)(A) Describe and perform transformations of figures in a plane using coordinate notation. (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (3)(B) Determine the image or pre-image of a given two-dimensional figures under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane.

A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar. A scale factor describes how much the figure is enlarged or reduced. For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b)  (ka, kb).

If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. The scale factor in fraction form is an improper fraction. If the scale factor is less than 1 (k < 1), it is a reduction. The scale factor in fraction form is a proper fraction. Helpful Hint

Example #1: Enlargement or reduction Example #1: Enlargement or reduction? Find the scale factor of each dilation. A. B. Each side = 4 in. Each side = 12 cm. Each side = 6 in. Each side = 9 cm. Reduction; scale factor = 9/12 = 3/4. Enlargement; scale factor = 6/4 = 3/2.

Example #2 Find the coordinates of the border of the photo after a dilation with scale factor Multiply all (x, y) by

Example #2 continued Plot the points A’(0, 0), B’(0, 10), C’(7.5, 10), D’(7.5, 0). No graph provided on notes, so it is not necessary to graph.

Example #3 Given that ∆TUO ~ ∆RSO, find the coordinates of U and the scale factor. Remember that distance is positive. Since ∆TUO ~ ∆RSO, Substitute 12 for RO, 9 for TO, and 16 for OY. 12(OU )= 144 Cross Products Prop. OU = 12 Divide both sides by 12. The coordinates of U are (0, 12). So the scale factor is

Example #4 Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2), and J(6, 2). What is the scale factor that maps ∆EHJ to ∆EFG? Which side of the triangles would be the easiest to use to write a ratio? This dilation is a reduction. HJ = 10 and FG = 5. Therefore the scale factor is 5/10 = ½.

Example #5 This is an enlargement. ST = 3 and UV = 9. Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and V(4, 3). What is the scale factor that maps ∆RST to ∆RUV? R S T U V This is an enlargement. ST = 3 and UV = 9. The scale factor is 9/3 = 3.

Example #6 Graph the image of ∆ABC after a dilation with scale factor Multiply each coordinate by the scale factor. A’ (0, 2) B’ (2, 4) C’ (4, 0)

Example #7 Graph the image of ∆MNP after a dilation with scale factor 3. M’(-6, 3) N’(6, 6) P’(-3, -3)

Example #8