[G.SRT.1.a]

First, let's calculate the slopes of the line segments in our preimage.

Our equation for the slope of a line is

m = y₂ - y₁

x₂ - x₁

where x₁ and y₁ are the x and y coordinates of the first point and x₂ and y₂ are the x and y coordinates of the second point. Note that it doesn't matter which point you use as the first or second, you'll get the same answer

For line segment BA, let's let B be the first point and A be the second point (we would get the same answer even if we switch these)

B = (-2,1), so x₁ = -2 and y₁ = 1

A = (-1,-1), so x₂ = -1 and y₂ = -1

Plugging these in to our slope equation

m = -1 - 1 = -2 = -2

-1 - (-2) 1

Performing the same calculation for the other line segments, we get the following results

Objective

Common Core Text:

- [G.SRT.1] Verify experimentally the properties of dilations given by a center and a scale factor:
- [G.SRT.1.a] A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

Said Differently:

- For dilations, show that
  - Lines of the image are parallel to lines of the pre-image
  - Lines passing through the center of dilation are unchanged

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Explanation

Let's see what happens to the slopes of lines when a dilation takes place

Example 1

Dilate the following shape by a factor of 2 about the origin. Describe the relationship between the slopes of the line segments in the preimage and image.

Line Segment

Slope

-2

1/3

Vertical line

Calculating the slopes of the line segments of the image,

Now let's perform the dilation.

Line Segment

Slope

-2

1/3

Vertical line

Compare results

Line Segment

Slope

-2

1/3

Vertical Line

Line Segment

Slope

-2

1/3

Vertical line

As you can see, the slopes of the line segments in the image and preimage are the same. Lines with equal slopes are parallel, so we could also say that the lines in the image and preimage are parallel. The one special case would be line DA. This line crosses through the center of dilation, so image and preimage are right on top of each other. (In my opinion, it's not a special case. They're still two parallel lines, who cares if they're on top of each other.)

In summary, we've found that, in a dilation

- corresponding lines and line segments of the image and preimage are parallel, except
  - lines (not line segments!) passing through the center of dilation. They remain unchanged.

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