Question
The blue figure is a dilation image of the black figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation.
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A graph has a horizontal x-axis from 0 to 16 in intervals of 1 and a vertical y-axis from 0 to 16 in intervals of 1. A black square and a blue square are plotted. The blue square has vertices at the following points: (2, 2); (5, 2); (2, 5); (5, 5). The black square has the following vertices: (6, 6); (6, 15); (15, 15); (15, 6).
Question content area bottom
Part 1
Is the dilation an enlargement or a reduction?
Enlargement
Reduction
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A graph has a horizontal x-axis from 0 to 16 in intervals of 1 and a vertical y-axis from 0 to 16 in intervals of 1. A black square and a blue square are plotted. The blue square has vertices at the following points: (2, 2); (5, 2); (2, 5); (5, 5). The black square has the following vertices: (6, 6); (6, 15); (15, 15); (15, 6).
Question content area bottom
Part 1
Is the dilation an enlargement or a reduction?
Enlargement
Reduction
Answers
GPT-4o mini
To determine whether the dilation is an enlargement or a reduction, we need to compare the sizes of the black and blue figures.
The blue square has vertices at:
- (2, 2)
- (5, 2)
- (2, 5)
- (5, 5)
The side length of the blue square is:
\[ 5 - 2 = 3 \]
Thus, the area of the blue square is:
\[ 3 \times 3 = 9 \]
The black square has vertices at:
- (6, 6)
- (6, 15)
- (15, 15)
- (15, 6)
The side length of the black square is:
\[ 15 - 6 = 9 \]
Thus, the area of the black square is:
\[ 9 \times 9 = 81 \]
### Step 1: Determine if it's an enlargement or reduction
Since the area of the blue square (9) is smaller than the area of the black square (81), the dilation from the black square to the blue square is a **reduction**.
### Step 2: Find the scale factor
The scale factor of dilation can be found by comparing the side lengths of the two squares. The scale factor \( k \) is given by the ratio of the side lengths of the blue square to the black square.
- Side length of the blue square: \( 3 \)
- Side length of the black square: \( 9 \)
The scale factor is calculated as:
\[ k = \frac{\text{Side length of blue square}}{\text{Side length of black square}} = \frac{3}{9} = \frac{1}{3} \]
Thus, the scale factor of the dilation is \( \frac{1}{3} \).
### Summary:
- The dilation is a **reduction**.
- The scale factor is \( \frac{1}{3} \).
The blue square has vertices at:
- (2, 2)
- (5, 2)
- (2, 5)
- (5, 5)
The side length of the blue square is:
\[ 5 - 2 = 3 \]
Thus, the area of the blue square is:
\[ 3 \times 3 = 9 \]
The black square has vertices at:
- (6, 6)
- (6, 15)
- (15, 15)
- (15, 6)
The side length of the black square is:
\[ 15 - 6 = 9 \]
Thus, the area of the black square is:
\[ 9 \times 9 = 81 \]
### Step 1: Determine if it's an enlargement or reduction
Since the area of the blue square (9) is smaller than the area of the black square (81), the dilation from the black square to the blue square is a **reduction**.
### Step 2: Find the scale factor
The scale factor of dilation can be found by comparing the side lengths of the two squares. The scale factor \( k \) is given by the ratio of the side lengths of the blue square to the black square.
- Side length of the blue square: \( 3 \)
- Side length of the black square: \( 9 \)
The scale factor is calculated as:
\[ k = \frac{\text{Side length of blue square}}{\text{Side length of black square}} = \frac{3}{9} = \frac{1}{3} \]
Thus, the scale factor of the dilation is \( \frac{1}{3} \).
### Summary:
- The dilation is a **reduction**.
- The scale factor is \( \frac{1}{3} \).