dilation

What is dilation in math? Definition.

Dilation is a mathematical transformation that changes the size of a figure without altering its shape. It involves stretching or shrinking the figure by a certain scale factor. This transformation is commonly used in geometry to resize shapes while maintaining their proportions.

History of dilation.

The concept of dilation can be traced back to ancient Greece, where mathematicians like Euclid and Archimedes studied the properties of similar figures. However, the formal definition and study of dilation as a transformation began in the 19th century with the development of projective geometry.

What grade level is dilation for?

Dilation is typically introduced in middle school mathematics, around grades 6-8. It serves as an important topic in geometry and lays the foundation for more advanced concepts like similarity and transformations.

What knowledge points does dilation contain? And detailed explanation step by step.

Dilation involves several key concepts and knowledge points:

1. Scale factor: The scale factor determines how much the figure will be stretched or shrunk. It is a positive real number greater than zero.

2. Center of dilation: This is the fixed point around which the figure is dilated. All points in the figure are stretched or shrunk away from or towards this center.

3. Similarity: Dilation preserves the shape of the figure, making it similar to the original. The corresponding angles remain congruent, and the corresponding sides are proportional.

The step-by-step process of dilation can be summarized as follows:

1. Identify the center of dilation and the scale factor.

2. Draw lines connecting each vertex of the figure to the center of dilation.

3. Measure the distance from each vertex to the center of dilation.

4. Multiply each distance by the scale factor to determine the new distances.

5. Use the new distances to plot the dilated figure.

Types of dilation.

There are two types of dilation:

1. Enlargement: In an enlargement, the scale factor is greater than 1, resulting in a larger figure.

2. Reduction: In a reduction, the scale factor is between 0 and 1, resulting in a smaller figure.

Properties of dilation.

Dilation exhibits the following properties:

1. The center of dilation, scale factor, and orientation determine a unique dilation.

2. Dilation preserves the shape of the figure, making it similar to the original.

3. The ratio of corresponding lengths in the original figure and the dilated figure is equal to the scale factor.

4. The ratio of corresponding areas in the original figure and the dilated figure is equal to the square of the scale factor.

How to find or calculate dilation?

To find or calculate dilation, follow these steps:

1. Identify the center of dilation and the scale factor.

2. Measure the distance from each vertex of the figure to the center of dilation.

3. Multiply each distance by the scale factor to determine the new distances.

4. Use the new distances to plot the dilated figure.

What is the formula or equation for dilation?

The formula for dilation can be expressed as:

(x', y') = (k * x, k * y)

where (x, y) represents the coordinates of a point in the original figure, (x', y') represents the coordinates of the corresponding point in the dilated figure, and k represents the scale factor.

How to apply the dilation formula or equation?

To apply the dilation formula, substitute the coordinates of each point in the original figure into the equation and multiply them by the scale factor. This will give you the coordinates of the corresponding points in the dilated figure.

What is the symbol or abbreviation for dilation?

The symbol for dilation is a capital letter "D" with an arrow on top, pointing towards the dilated figure. It is often written as "Dil."

What are the methods for dilation?

There are several methods for dilation:

1. Using a compass and ruler: This traditional method involves physically measuring and marking the distances to dilate the figure.

2. Using coordinate geometry: This method involves using the dilation formula to calculate the coordinates of the dilated figure.

3. Using technology: With the advent of computer software and graphing calculators, dilation can be easily performed using digital tools.

More than 3 solved examples on dilation.

Example 1: Dilate triangle ABC with a scale factor of 2 and center of dilation at point D. The coordinates of the vertices are A(2, 3), B(4, 5), and C(6, 1).

Solution:

1. Measure the distance from each vertex to point D:

   • AD = √[(2-0)^2 + (3-0)^2] = √13
   • BD = √[(4-0)^2 + (5-0)^2] = √41
   • CD = √[(6-0)^2 + (1-0)^2] = √37

2. Multiply each distance by the scale factor:

   • AD' = 2 * √13 = 2√13
   • BD' = 2 * √41 = 2√41
   • CD' = 2 * √37 = 2√37

3. Plot the dilated triangle with the new distances:

   • A'(2√13, 3√13)
   • B'(2√41, 2√41)
   • C'(2√37, 2√37)

Example 2: Dilate rectangle PQRS with a scale factor of 0.5 and center of dilation at point O. The coordinates of the vertices are P(2, 4), Q(6, 4), R(6, 2), and S(2, 2).

Solution:

1. Measure the distance from each vertex to point O:

   • OP = √[(2-4)^2 + (4-4)^2] = 2
   • OQ = √[(6-4)^2 + (4-4)^2] = 2
   • OR = √[(6-4)^2 + (2-4)^2] = 2
   • OS = √[(2-4)^2 + (2-4)^2] = 2

2. Multiply each distance by the scale factor:

   • OP' = 0.5 * 2 = 1
   • OQ' = 0.5 * 2 = 1
   • OR' = 0.5 * 2 = 1
   • OS' = 0.5 * 2 = 1

3. Plot the dilated rectangle with the new distances:

   • P'(1, 4)
   • Q'(3, 4)
   • R'(3, 3)
   • S'(1, 3)

Example 3: Dilate circle O with a scale factor of 3 and center of dilation at point A. The equation of the circle is (x-2)^2 + (y-3)^2 = 4.

Solution:

1. Measure the distance from the center of the circle to point A:

   • OA = √[(2-0)^2 + (3-0)^2] = √13

2. Multiply the distance by the scale factor:

   • OA' = 3 * √13 = 3√13

3. Substitute the new distance into the equation of the circle:

   • (x-2)^2 + (y-3)^2 = (3√13)^2
   • (x-2)^2 + (y-3)^2 = 117

Practice Problems on dilation.

1. Dilate triangle XYZ with a scale factor of 0.75 and center of dilation at point B. The coordinates of the vertices are X(1, 2), Y(3, 4), and Z(5, 2). Find the coordinates of the dilated triangle.

2. Dilate rectangle ABCD with a scale factor of 2.5 and center of dilation at point O. The coordinates of the vertices are A(1, 1), B(1, 4), C(4, 4), and D(4, 1). Find the coordinates of the dilated rectangle.

3. Dilate circle O with a scale factor of 0.6 and center of dilation at point A. The equation of the circle is (x+2)^2 + (y-1)^2 = 9. Find the equation of the dilated circle.

FAQ on dilation.

Question: What is dilation?

Answer: Dilation is a mathematical transformation that changes the size of a figure without altering its shape. It involves stretching or shrinking the figure by a certain scale factor.

Question: What is the difference between enlargement and reduction in dilation?

Answer: Enlargement is a dilation where the scale factor is greater than 1, resulting in a larger figure. Reduction is a dilation where the scale factor is between 0 and 1, resulting in a smaller figure.

Question: How does dilation preserve the shape of a figure?

Answer: Dilation preserves the shape of a figure by maintaining the congruence of corresponding angles and the proportionality of corresponding sides. This ensures that the dilated figure is similar to the original.

Question: Can dilation change the orientation of a figure?

Answer: No, dilation does not change the orientation of a figure. The dilated figure will have the same orientation as the original.

Question: Can dilation change the position of a figure?

Answer: Yes, dilation can change the position of a figure if the center of dilation is not the same as the center of the original figure. The dilated figure will be shifted towards or away from the center of dilation.

Question: Can dilation be performed on three-dimensional objects?

Answer: Yes, dilation can be performed on three-dimensional objects as well. The same principles of scale factor and center of dilation apply, but the calculations and visualizations become more complex.