Think About the Process The graph shows Upper DeltaJKL and Upper Delta​J'K'L', its image after a dilation. Is this dilation an enlargement or a​ reduction? Explain. Then find the scale factor of the dilation.

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Part 1
Is the dilation an enlargement or a​ reduction? Explain.
A.
A​ reduction, because the image figure is larger than the original figure.
B.
An​ enlargement, because the image figure is smaller than the original figure.
C.
An​ enlargement, because the image figure is larger than the original figure.
D.
A​ reduction, because the image figure is smaller than the original figure.

Answer 1

To determine whether the dilation is an enlargement or a reduction, we need to analyze the relationship between the original triangle \( \Delta JKL \) and its image \( \Delta J'K'L' \).

1. Enlargement vs. Reduction:
   • Enlargement: If the image is larger than the original figure, it is called an enlargement.
   • Reduction: If the image is smaller than the original figure, it is called a reduction.

Given the options:

• A. A reduction, because the image figure is larger than the original figure.
• B. An enlargement, because the image figure is smaller than the original figure.
• C. An enlargement, because the image figure is larger than the original figure.
• D. A reduction, because the image figure is smaller than the original figure.

Based on your observation of the graph:

• If the vertices \( J', K', L' \) are farther from the origin (or center of dilation) than \( J, K, L \), then it's an enlargement.
• If \( J', K', L' \) are closer to the origin than \( J, K, L \), then it's a reduction.

Assuming that the vertices \( J', K', L' \) are smaller and closer to the center of dilation than \( J, K, L \), the correct answer would be:

D. A reduction, because the image figure is smaller than the original figure.

2. Finding the Scale Factor: To find the scale factor of the dilation, use the coordinates of the vertices of \( \Delta JKL \) and \( \Delta J'K'L' \):

• If we denote the vertices of \( J, K, L \) as (x1, y1), (x2, y2), (x3, y3) and the vertices of \( J', K', L' \) as (x1', y1'), (x2', y2'), (x3', y3'), the scale factor \( k \) can be calculated as:

\[ k = \frac{\text{Distance from center to image}}{\text{Distance from center to original}} \]

Choose a point (like the origin (0,0) or another point, depending on your graph) as the center of dilation to find the scale factor.

Calculate the length of the sides of both triangles and determine the ratio. If you can provide specific coordinates, I can help you find the exact scale factor.