Question

activity 2: dilation obstacle course
here is a diagram that shows nine points.

1. dilate b using a scale factor of 5 and a as the center of dilation. which point is its image?
2. using h as the center of dilation, dilate g so that its image is e. what scale factor did you use?
3. using h as the center of dilation, dilate e so that its image is g. what scale factor did you use?
4. to dilate f so that its image is b, what point on the diagram could you use as a center?
5. dilate h using a as the center and a scale factor of 1/3. which point is its image?
6. describe a dilation that uses a labeled point as its center and that would take f to h.
7. using b as the center of dilation, dilate h so that its image is itself. what scale factor did you use?

Explanation:

Step1: Understanding Dilation

Dilation is a transformation that changes the size of a figure. If the center of dilation is \(C\) and the scale factor is \(k\), then for a point \(P\), its image \(P'\) is such that \(\overrightarrow{CP'}=k\overrightarrow{CP}\)

Step2: Solving Problem 1

Let's assume we can count the "distance" (in terms of the grid - like structure implied by the points) from \(A\) to \(B\). If we consider the relative position of the points, when we dilate \(B\) with center \(A\) and scale factor \(k = 5\).
If we assume the "unit" distance from \(A\) to \(B\) is \(1\) unit (in the context of the point - to - point spacing in the diagram), then \(\overrightarrow{AB}\) is extended. The image of \(B\) after dilation with center \(A\) and scale factor \(5\) is \(C\)

Step3: Solving Problem 2

Let \(H\) be the center of dilation. Let \(\overrightarrow{HG}\) and \(\overrightarrow{HE}\). If we assume the "distance" from \(H\) to \(G\) is \(2\) units and from \(H\) to \(E\) is \(1\) unit (by counting the relative positions of the points), and using the formula \(P'=kP\) (with center \(H\)), \(\overrightarrow{HE}=k\overrightarrow{HG}\). So \(k=\frac{1}{2}\)

Step4: Solving Problem 3

Using \(H\) as the center of dilation. Let \(\overrightarrow{HE}\) and \(\overrightarrow{HG}\). Since \(\overrightarrow{HG}=k\overrightarrow{HE}\), and if \(\overrightarrow{HE}\) is \(1\) unit and \(\overrightarrow{HG}\) is \(2\) units (by relative position), then \(k = 2\)

Step5: Solving Problem 4

Let's check the lines of points. If we consider the line passing through \(F\) and \(B\). If we take \(D\) as the center of dilation. Let the "distance" from \(D\) to \(F\) be \(1\) unit and from \(D\) to \(B\) be \(2\) units (by relative position), then a dilation with center \(D\) can map \(F\) to \(B\)

Step6: Solving Problem 5

Dilate \(H\) with center \(A\) and scale factor \(k=\frac{1}{3}\). If we assume the "distance" from \(A\) to \(H\) is \(3\) units (by relative position of the points), then the image of \(H\) is \(D\)

Step7: Solving Problem 6

Let's take \(G\) as the center of dilation. Let the "distance" from \(G\) to \(F\) be \(1\) unit and from \(G\) to \(H\) be \(2\) units (by relative position). A dilation with center \(G\) and scale factor \(k = 2\) will map \(F\) to \(H\)

Step8: Solving Problem 7

When a point is its own image under dilation, the scale factor \(k = 1\). Because \(\overrightarrow{BH}=k\overrightarrow{BH}\), and the only value of \(k\) for which \(P'=P\) (where \(P = H\) and center of dilation is \(B\)) is \(k = 1\)

Answer:

1. \(C\)
2. \(\frac{1}{2}\)
3. \(2\)
4. \(D\)
5. \(D\)
6. Dilate \(F\) with center \(G\) and scale factor \(2\)
7. \(1\)