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Chapter 9: Problem 12

Find the measure of the dilation image or the preimage of \(\overline{S T}\) using the given scale factor. \(S^{\prime} T^{\prime}=12, r=\frac{2}{3}\)

Short Answer

Expert verified
The preimage length \(\overline{ST}\) is 18 units.

Step by step solution

01

Understand the Problem

We need to find the measure of the dilation preimage, \(\overline{ST}\), given that its image, \(\overline{S'T'}\), is 12 units long with a scale factor \(r\) of \(\frac{2}{3}\).
02

Use the Formula for Dilation

The formula to find the preimage length \(ST\) when the image length \(S'T'\) and the scale factor \(r\) are known is \(ST = \frac{S'T'}{r}\).
03

Substitute the Known Values

Plug the known values into the formula: \(ST = \frac{12}{\frac{2}{3}}\).
04

Simplify the Expression

To divide by a fraction, multiply by its reciprocal. So, \(ST = 12 \times \frac{3}{2}\).
05

Calculate the Preimage Length

Calculate \(12 \times \frac{3}{2} = 18\). Therefore, the length of the preimage \(\overline{ST}\) is 18 units.

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Key Concepts

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

Understanding Scale Factor in Dilation
In geometry, dilation is a transformation that changes the size of a figure but not its shape. A key component of dilation is the **scale factor**. The scale factor determines how much the size changes. It tells us whether the figure becomes larger or smaller and by what proportion.

- If the scale factor is greater than 1, the image gets larger; it's an enlargement.- If the scale factor is less than 1, the image gets smaller; it's a reduction.
In our exercise, the scale factor is given as \( r = \frac{2}{3} \). This indicates that our dilation will result in a reduction since \( \frac{2}{3} \) is less than 1, meaning each point of the image is only \( \frac{2}{3} \) the distance from the center of dilation compared to points on the preimage.
Exploring Preimage and Image Relationships
In dilation, the **preimage** is the original figure, and the **image** is the transformed figure. Understanding these terms is essential because they help distinguish what you're calculating.

- **Preimage**: The original shape or length before dilation.- **Image**: The resulting shape or length after dilation.
The exercise states that the image \( \overline{S'T'} \) is 12 units long. This image results from a dilation applied to the unknown preimage \( \overline{ST} \). The task is to find this original preimage length using the given scale factor.
Calculating Length in Dilation
To find the length of a preimage or image in dilation, we use a formula that relates the lengths with the scale factor. For this exercise, we are given the image length \( S'T' = 12 \) and asked to find the preimage length \( ST \).

The formula: \[ ST = \frac{S'T'}{r} \]This formula tells us how to work backwards from the image length to calculate the preimage length using the scale factor.

Let's apply this knowledge: Substituting in the known values, we have \( ST = \frac{12}{\frac{2}{3}} \). Dividing by a fraction is the same as multiplying by its reciprocal, so it becomes \( ST = 12 \times \frac{3}{2} \).

Finally, calculating this gives us \( ST = 18 \). This means the preimage \( \overline{ST} \) is 18 units in length, explaining the shorter image length due to the reduction by the scale factor.

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Most popular questions from this chapter

Grace is creating a template for the class newsletter. She has a photograph that is 10 centimeters by 12 centimeters, but the maximum space available for the photograph is 6 centimeters by 8 centimeters. She wants the photograph to be as large as possible on the page. When she uses a scanner to save the photograph, at what percent of the original photograph's size should she save the image file?

Find the measure of the dilation image or the preimage of \(\overline{A B}\) with the given scale factor. \(A^{\prime} B^{\prime}=15, r=3\)

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

Find the magnitude and direction of each resultant for the given vectors. \(\overrightarrow{\mathbf{y}}=\langle 9,-10\rangle, \overrightarrow{\mathbf{z}}=\langle- 10,-2\rangle\)

Find the distance between each pair of parallel lines. $$\begin{aligned}&x=-2\\\&x=5\end{aligned}$$
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