If ABCD were dilated by a scale factor of with the center of dilation being the 2 origin, describe the effects on ABCD.

Question

Be sure to address side lengths, angle measures, and relative location. If ABCD were dilated by a scale factor of 2, how would your answer change?

A = (4,4)

B = (8,4)

C = (7,7)

E =(5,7)

Answer 1

When a shape, such as quadrilateral ABCD, is dilated from a center of dilation (in this case, the origin \((0, 0)\)), the effects of the dilation relate to the scale factor being applied. Let's analyze what happens under dilation with different scale factors.

General Effects of Dilation:

1. Side Lengths: The side lengths of the figure will be multiplied by the scale factor. If the scale factor is \( k \), then each side length will become \( k \times \text{(original length)} \).

2. Angle Measures: The angles in the figure will remain unchanged. Dilation does not affect the angle measures of shapes.

3. Relative Location: The relative positions of all points in the figure will change according to the center of dilation. Each point will be moved away from the center by a factor of the scale factor. The new coordinates \((x', y')\) of each original point \((x, y)\) under dilation from the origin can be calculated as:

   \[ (x', y') = (k \times x, k \times y) \]

Dilation with a Scale Factor of 1:

If quadrilateral ABCD is dilated by a scale factor of \( 1 \), the effects would be:

1. Side Lengths: Each side length remains unchanged.
2. Angle Measures: All angle measures remain unchanged.
3. Relative Location: The location of all points does not change; they stay the same.

Dilation with a Scale Factor of 2:

Now, let's consider the effects when ABCD is dilated by a scale factor of \(2\):

• A = (4, 4) becomes: \[ (2 \times 4, 2 \times 4) = (8, 8) \]

• B = (8, 4) becomes: \[ (2 \times 8, 2 \times 4) = (16, 8) \]

• C = (7, 7) becomes: \[ (2 \times 7, 2 \times 7) = (14, 14) \]

• D = (5, 7) becomes: \[ (2 \times 5, 2 \times 7) = (10, 14) \]

Summary of Dilation by a Scale Factor of 2:

1. New Coordinates:

   • A' = (8, 8)
   • B' = (16, 8)
   • C' = (14, 14)
   • D' = (10, 14)

2. Side Lengths: The lengths of each side of quadrilateral ABCD will be doubled.

3. Angle Measures: The angle measures will remain the same as in the original quadrilateral.

4. Relative Location: The entire shape is moved away from the origin (0, 0) and is scaled up, with the new coordinates being proportionally farther from the origin than the original points.

Therefore, the dilated figure A'B'C'D' is similar to the original ABCD, and maintains the same shape but is larger by a factor of \( 2 \).