#### Need Help?

Get in touch with us

# Dilations in Geometry

## 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

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […] #### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […] #### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]   