Dilations
Lesson 14.4
Pre-AP Geometry
Lesson Focus
There are transformations that do not preserve distance.
This lesson introduces one such transformation, called a
dilation.
Basic Terms
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 OP and OP’ = k · OP.
2) If k < 0, P’ lies on the ray opposite OP and OP’ = |k| · OP.
Basic Terms
Expansion
A dilation where the scale factor |k| > 1.
Contraction
A dilation where the scale factor |k| < 1.
Basic Terms
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.
Theorem 14-5
A dilation maps a triangle to a similar triangle.
Corollary 1
A dilation maps an angle to a congruent angle.
Corollary 2
A dilation DO,k maps any segment to a parallel segment |k|
times as long.
Corollary 3
A dilation DO,k maps any polygon to a similar polygon whose
area is k2 times as large.
Practice
1.
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?
Review
1. 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-axis
b. the y-axis
c. the line y = x
Written Exercises
Problem Set 14.4, p.590: # 2 - 22 (even);
Handout 14-4