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# Dilations in Geometry

Sep 13, 2022

## 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:

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.

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 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 3:

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

### 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.

### 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’.

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 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”

## 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.
1. 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.
1. Find the scale factor. Tell whether the dilation is a reduction or an enlargement. Find the value of x.
1. Copy the diagram. Then draw the given dilation.

Center H; k = 2.

1. 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.

### 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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