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Dilations

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Dilations

Lesson 14.5

Pre-AP Geometry

There are transformations that do not preserve distance. This lesson introduces one such transformation, called a dilation.

Dilation

A transformation that changes the size of a figure by a scale factor to create a similar figure. A dilation is not rigid.

DO,k

A dilation with a center O and a nonzero scale factor k maps any point P to a point P’.

1) If k > 0, P’ lies on and OP’ = k · OP.

2) If k < 0, P’ lies on the ray opposite and OP’ = |k| · OP.

Expansion

A dilation where the scale factor |k| > 1.

Contraction

A dilation where the scale factor |k| < 1.

Similarity Mapping

A transformation that maps any geometric figure to a similar geometric figure.

A dilation is not an isometry as distances are not preserved.

A dilation maps a triangle to a similar triangle.

A dilation maps an angle to a congruent angle.

A dilation DO,k maps any segment to a parallel segment |k| times as long.

A dilation DO,k maps any polygon to a similar polygon whose area is k2 times as large.

- Given: A(3, 6), B(-3, -3), and C(-6, 0).
Find: (a) DO,2 ; (b) DO, -1/3

2. A dilation with the origin, O, as center maps (3, 4) to (9, 12). Find the scale factor. Is the dilation an expansion or a contraction?

- A dilation with the origin, O, as center maps
(-3, 4) → (1, -4/3). Find the scale factor.

Is the dilation an expansion or contraction?

- Which transformations are isometries?
- If g(x) = 7 – 2x, find the image of 3 and the preimage of -5.
- If R: (x, y) → (x – 2, y + 3), find the image of (-3, 1).
- Find the image of (-1, 4) when reflected in each line.
a. the x-axis

b. the y-axis

c. the line y = x

Problem Set 14.5, p. 596: # 2 – 8 (even)

Handout: 14-5