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Dilations in Geometry

Sep 13, 2022
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Key Concepts

  • Identify dilations
  • Draw a dilation
  • Understand scalar multiplication
  • Use scalar multiplications in a dilation
  • To find the image of a composition

Dilation

A dilation is a transformation in which the original figure and its image are similar. 

A dilation with center C and scale factor k maps every point P in a figure to a point P’ so that one of the following statements is true: 

If P is not the center point C, then the image point P’ lies on CP− The scale factor k is a positive number such that k=CP’/ CP and k ≠ 1, or 

If P is the center point C, then P = P’ 

Note: 

parallel

You can describe a dilation with respect to the origin with the notation 

(x, y) → (kx, ky), where k is the scale factor. 

The dilation is a reduction if 0 < k < 1 and it is an enlargement if k > 1. 

Identify Dilation

Example: 

Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. 

parallel
Identify Dilation: 

Solution: 

Here,

k = CP’/ CP = 12 / 8 = 3 / 2

The scale factor is 3 / 2>1

The image P’ is an enlargement. 

Draw a Dilation

Let us understand this concept with the help of an example: 

Draw and label parallelogram DEFG. Then construct a dilation of parallelogram DEFG with point D as the center of dilation and a scale factor of 2. 

Solution: 

Step 1: 

Draw DEFG. Draw rays from D through vertices E, F, and G. 

Step: 1 

Step 2: 

Open the compass to the length of DE− Locate E’ on DE so 

DE’ = 2(DE). Locate F’ and G’ the same way. 

Step: 2 

Step 3: 

Add a second label D’ to point D. Draw the sides of D’E’F’G’. 

Step: 3 

Scalar multiplication

Scalar multiplication is the process of multiplying each element of a matrix by a real number or scalar. 

Example: 

Dilations using matrices

You can use scalar multiplication to represent a dilation centered at the origin in the coordinate plane. To find the image matrix for a dilation centered at the origin, use the scale factor as the scalar. 

Use scalar multiplication in dilations

Let us understand this concept with the help of an example: 

The vertices of quadrilateral ABCD are A(-6, 6), B(-3, 6), C(0, 3), and D(-6, 0). Use scalar multiplication to find the image of ABCD after a dilation with its center at the origin and a scale factor of  1/ 3 Graph ABCD and its image. 

Dilations using matrices: 

Find the image of the composition

The vertices of ∆ ABC are A(-4, 1), B(-2, 2), and C(-2, 1). Find the image of ∆ ABC after the given composition. 

Translation: (x, y) → (x + 5, y + 1)  

Dilation: Centered at the origin with a scale factor of 2. 

Solution: 

STEP 1: Graph the preimage ∆ABC on the coordinate plane. 

STEP 2: Translate ∆ABC 5 units to the right and 1 unit up. Label it ΔA’B’C’. 

STEP 3: Dilate ∆ A’B’C’ using the origin as the center and a scale factor of 2 to find ∆ A’B’C’. 

Solution: 

Example: 

1. A segment has the endpoints A(-1, 1) and B(1, 1). Find the image AB−AB- after a 90°° rotation about the origin followed by dilation with its center at the origin and a scale factor of 2. 

Solution: 

The given line segment has endpoints A(-1, 1) and B(1, 1). Graph line

AB Rotation of 90° about the origin: 

For a rotation of 90°

(a, b) → (-b, a)

A(-1, 1) → A’(-1,-1) 

B(1, 1) → B’(-1,1) 

Now lets graph AB

Now lets graph AB

Now we need to perform dilation with the center as the origin and scale factor 2. 

Dilation rule here, 

(x, y) à (2x, 2y) 

A’(-1, -1) à A”(-2, -2) 

B’(-1, 1)à B”(-2, 2) 

So, Let us graph line A”B”

So, Let us graph line A''B''

Exercise

  1. Find the coordinates of A, B, and C so that ABC is a dilation of PQR with a scale factor of k. Sketch PQR and ABC. P(-2, -1), Q(-1, 0), R(0, -1); k = 4
  2. A triangle has the vertices A(4, -4), B(8, 2), and C(8, -4). The image of ABC after a dilation with a scale factor of  is DEF. Sketch ABC and DEF.
  3. Draw a dilation of quadrilateral ABCD with vertices A(2, 1), B(4, 1), C(4, -1), and D(1, -1). Use a scale factor of 2.
  4. Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find its scale factor.
Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find its scale factor.
  1. Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find its scale factor.
Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find its scale factor.
  1. Find the scale factor. Tell whether the dilation is a reduction or an enlargement. Find the value of x.
Find the scale factor. Tell whether the dilation is a reduction or an enlargement. Find the value of x.
  1. Find the scale factor. Tell whether the dilation is a reduction or an enlargement. Find the value of x.
7
  1. Copy the diagram. Then draw the given dilation.
Copy the diagram. Then draw the given dilation.

Center H; k = 2.

  1. Simplify the product.
Simplify the product.
  1. Find the image matrix that represents a dilation of the polygon centered at the origin with the given scale factor. Then graph the polygon and its image.
Find the image matrix that represents a dilation of the polygon centered at the origin with the given scale factor. Then graph the polygon and its image.

Concept Map

Concept Map

What we have learned

  • Identify dilations
  • Draw a dilation
  • Understand scalar multiplication
  • Use scalar multiplications in a dilation
  • To find the image of a composition

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