Question
upstudy study bank question image url

Which of these properties are not preserved by a dilation?

Ask by Campbell Lang. in the United States
Jan 24,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

Dilation preserves shape and angles but changes lengths and areas.

Solution

A **dilation** is a type of similarity transformation in geometry that changes the size of an object but preserves its shape. Specifically, dilation involves scaling all distances from a fixed center point by a common factor. While dilation maintains certain geometric properties, it does not preserve others. **Properties Preserved by Dilation:** - **Shape:** The overall form of the object remains unchanged. - **Angles:** All angles within the object remain the same. - **Parallelism:** Lines that are parallel before dilation remain parallel after dilation. - **Ratio of Distances:** The proportion between different distances within the object remains consistent. **Properties **Not** Preserved by Dilation:** - **Lengths:** Actual distances between points are scaled by the dilation factor, so absolute lengths change. - **Area:** The area of the object changes by the square of the dilation factor. - **Volume (in 3D):** The volume changes by the cube of the dilation factor. **Example:** If you have a triangle and apply a dilation with a scaling factor of 2: - The shape and angles of the triangle remain the same. - Each side length doubles. - The area of the triangle increases by a factor of 4. **Conclusion:** While dilation maintains the shape and angular properties of geometric figures, it does not preserve the actual sizes, such as lengths and areas. **Answer:** Absolute sizes like lengths and areas change under dilation, so lengths and areas are not preserved.

Reviewed and approved by the UpStudy tutoring team

error msg
Explain
Simplify this solution

The Deep Dive

Dilation is a transformation that alters the size of a figure without changing its shape. While the angles of the figure remain the same in a dilation, the actual lengths of the sides do not. This means that properties like distance and perimeter are not preserved during dilation. So, while angles stay constant, any measurements that involve distance will change based on the scale factor used. Dilation can be applied in various real-world scenarios, such as in architecture and graphic design. For instance, when creating scaled models of buildings, architects use dilation to ensure that the proportions remain the same as the actual structure, allowing them to visualize and communicate design concepts effectively. This transformation is also essential in computer graphics for resizing images without losing their original shape.

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy