Dilation Worksheet With Answers: Boost Your Math Skills!
Table of Contents :
Dilation is an essential concept in geometry that helps students understand how shapes change size while maintaining their proportions. Whether you’re a student looking to enhance your understanding of dilation or a teacher seeking effective resources, a dilation worksheet can be a fantastic tool to boost your math skills. In this article, we’ll explore what dilation is, how it works, and provide a sample worksheet complete with answers to solidify your learning.
Understanding Dilation
Dilation is a transformation that alters the size of a figure while keeping its shape proportional. It is characterized by a center point (also called the center of dilation) and a scale factor. When a figure is dilated, every point on the figure moves away from or towards the center of dilation based on the scale factor.
Key Terms to Know
- Center of Dilation: The fixed point around which a figure is enlarged or reduced.
- Scale Factor: The ratio that describes how much the figure is enlarged (greater than 1) or reduced (less than 1).
- Image: The resulting figure after dilation.
Visual Representation
To better understand dilation, let's visualize a simple triangle and see how it is affected by different scale factors.
<table> <tr> <th>Scale Factor</th> <th>Effect on Triangle</th> </tr> <tr> <td>1 (no change)</td> <td>The triangle remains the same size.</td> </tr> <tr> <td>2 (enlargement)</td> <td>The triangle doubles in size, maintaining its shape.</td> </tr> <tr> <td>0.5 (reduction)</td> <td>The triangle shrinks to half its size, keeping its proportionality.</td> </tr> </table>
How Dilation Works
The formula for dilation can be expressed as follows:
For a point ( P(x, y) ), when dilated with a scale factor ( k ) and centered at point ( O(h, k) ), the coordinates of point ( P' ) (the image) can be calculated using:
[ P' = O + k(P - O) ]
This translates to:
[ P' = (h + k(x - h), k + k(y - k)) ]
Example of Dilation
Let's consider a point ( A(2, 3) ) that is dilated with a center of dilation at ( O(1, 1) ) and a scale factor of 2.
Using the formula, the coordinates of ( A' ) can be calculated:
- ( A' = (1 + 2(2 - 1), 1 + 2(3 - 1)) )
- ( A' = (1 + 2, 1 + 4) )
- ( A' = (3, 5) )
Dilation Worksheet
Now that we understand the fundamental concepts of dilation, let’s put your skills to the test with a worksheet. Try to solve the following problems to reinforce your understanding.
Problems
- Dilate the point ( B(3, 4) ) with a scale factor of 3 centered at ( O(0, 0) ).
- If the triangle with vertices ( C(1, 1) ), ( D(3, 1) ), and ( E(2, 3) ) is dilated with a scale factor of 0.5 and centered at ( O(0, 0) ), find the coordinates of the new vertices.
- Find the new coordinates of point ( F(-2, 3) ) when dilated with a center at ( O(1, 1) ) and a scale factor of 2.
- A square has vertices ( G(2, 2) ), ( H(2, 4) ), ( I(4, 4) ), and ( J(4, 2) ). If it is dilated with a scale factor of 1.5 centered at ( O(0, 0) ), what are the new vertices?
Answers
Here are the answers for the worksheet above:
-
For point ( B(3, 4) ):
- ( B' = (0 + 3(3 - 0), 0 + 3(4 - 0)) = (9, 12) )
-
For triangle ( C(1, 1) ), ( D(3, 1) ), and ( E(2, 3) ):
- ( C' = (0 + 0.5(1 - 0), 0 + 0.5(1 - 0)) = (0.5, 0.5) )
- ( D' = (0 + 0.5(3 - 0), 0 + 0.5(1 - 0)) = (1.5, 0.5) )
- ( E' = (0 + 0.5(2 - 0), 0 + 0.5(3 - 0)) = (1, 1.5) )
-
For point ( F(-2, 3) ):
- ( F' = (1 + 2(-2 - 1), 1 + 2(3 - 1)) = (-3, 5) )
-
For square ( G(2, 2) ), ( H(2, 4) ), ( I(4, 4) ), and ( J(4, 2) ):
- ( G' = (0 + 1.5(2 - 0), 0 + 1.5(2 - 0)) = (3, 3) )
- ( H' = (0 + 1.5(2 - 0), 0 + 1.5(4 - 0)) = (3, 6) )
- ( I' = (0 + 1.5(4 - 0), 0 + 1.5(4 - 0)) = (6, 6) )
- ( J' = (0 + 1.5(4 - 0), 0 + 1.5(2 - 0)) = (6, 3) )
Conclusion
Dilation is a powerful geometric transformation that opens up new avenues for understanding shapes and their properties. With the help of a dilation worksheet, students can practice and reinforce their learning while gaining confidence in their math skills. Remember that practice makes perfect, so keep exploring dilation and other geometry concepts! Happy studying! 📏📐✏️