Dilations

Lesson 14.5Pre-AP Geometry

Lesson Focus

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

Basic TermsDilation

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.

OPOP

Basic TermsExpansion

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

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

Basic TermsSimilarity Mapping

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

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

Theorem 14-5

A dilation maps a triangle to a similar triangle.

Corollary 1A dilation maps an angle to a congruent angle.

Corollary 2A dilation DO,k maps any segment to a parallel segment |k| times as long.

Corollary 3A dilation DO,k maps any polygon to a similar polygon whose area is k2 times as large.

Practice1. 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?

3. 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?

Review1. Which transformations are isometries?

2. If g(x) = 7 – 2x, find the image of 3 and the preimage of -5.

3. If R: (x, y) → (x – 2, y + 3), find the image of (-3, 1).

4. Find the image of (-1, 4) when reflected in each line.a. the x-axisb. the y-axisc. the line y = x

Written Exercises

Problem Set 14.5, p. 596: # 2 – 8 (even)Handout: 14-5