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Chapter 5: Problem 15

Find the measure of the dilation image or the preimage of \(\overline{S T}\) using the given scale factor. \(S T=2.25, r=0.4\)

Short Answer

Expert verified
The length of the dilation image is 0.9.

Step by step solution

01

Understand the Problem

We are given a segment \( \overline{S T} \) with a length of 2.25 and a scale factor \( r = 0.4 \). Our task is to find the length of the segment after dilation using this scale factor.
02

Apply Dilation Formula

To find the length of the dilation image of a segment, use the formula: \[ \text{Dilation Image Length} = \text{Original Length} \times \text{Scale Factor} \]Substitute the given values: \[ \text{Dilation Image Length} = 2.25 \times 0.4 \]
03

Perform the Calculation

Calculate the product of the original length and the scale factor:\[ 2.25 \times 0.4 = 0.9 \]Thus, the length of the dilation image of \( \overline{S T} \) is 0.9.
04

Conclusion

The length of the dilation image \( \overline{S T} \) after applying the scale factor \( r = 0.4 \) is 0.9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Scale Factor
In geometry, a scale factor is a number that scales, or multiplies, some quantity. It is the ratio of any two corresponding lengths in a geometric model or graph. Scale factors are crucial when you want to change the size of a figure but keep its shape the same.

To understand scale factor better:
  • If a scale factor is greater than 1, the figure enlarges, making all parts bigger but in the same proportion.
  • If a scale factor is less than 1, the figure shrinks, reducing all parts symmetrically.
Dilation uses a scale factor to create an image that is similar to the original; they are the same shape but different sizes. This means that the lengths are proportional due to the scale factor's influence. For instance, in the problem, with a scale factor of 0.4, the segment shrinks, leading to a smaller version compared to the original length.
The Concept of Dilation Image
A dilation image refers to the resulting figure after each point of the preimage is moved along a line from a certain fixed point, known as the center of dilation, and it is multiplied by a scale factor. This manipulation maintains the figure's original shape, ensuring the sides remain in proportion.

Key points about dilation images:
  • The distance between points in the dilation image is scaled by the scale factor.
  • A positive scale factor means the points move away from the center of dilation, while a negative one means they move towards it.
  • Dilation does not affect the angles, meaning the image is similar to the preimage.
In the exercise, the segment \( \overline{S T} \) has a dilation image of 0.9 when using a scale factor of 0.4, indicating a proportional reduction in length without altering the figure's shape.
Strategies for Solving Geometry Problems
Solving dilation problems in geometry requires a methodical approach. It's all about understanding the relationship between the original figure and its dilated image. Let's break down a structured strategy for problem-solving:

  • **Read & Identify:** Start by carefully reading the problem. Identify known values such as the original length and scale factor.
  • **Formula Application:** Use the appropriate formula, such as the dilation formula for finding image length: \[ \text{Dilation Image Length} = \text{Original Length} \times \text{Scale Factor} \]
  • **Calculations:** Perform accurate calculations after substituting the known values into the formula. Double-check results for mistakes.
  • **Verification:** Verify the solution by checking if the resultant image length aligns with the given scale situation.
In the given exercise, employing these strategies ensures accuracy and enhances understanding in transforming geometric figures using dilation.

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