Dilations
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Dilations. Lesson 14.5 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

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Dilations

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Dilations

Dilations

Lesson 14.5

Pre-AP Geometry


Lesson focus

Lesson Focus

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


Basic terms

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 and OP’ = k · OP.

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


Basic terms1

Basic Terms

Expansion

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

Contraction

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


Basic terms2

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

Theorem 14-5

A dilation maps a triangle to a similar triangle.


Corollary 1

Corollary 1

A dilation maps an angle to a congruent angle.


Corollary 2

Corollary 2

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


Corollary 3

Corollary 3

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


Practice

Practice

  • 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?


Review

Review

  • 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


Written exercises

Written Exercises

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

Handout: 14-5


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