Question

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 c 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 can you use as the center of dilation? 5. dilate h using a as the center of dilation and a scale factor of 1/3. which point is its image? 6. describe a dilation that uses a labeled point as its center of dilation 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: Recall dilation formula

If the center of dilation is \(O(x_0,y_0)\) and the point to be dilated is \(P(x,y)\) with a scale - factor \(k\), the image \(P'(x',y')\) is given by \(x'=x_0 + k(x - x_0)\) and \(y'=y_0 + k(y - y_0)\). In a geometric sense, if we have a center of dilation \(O\) and a point \(P\), the vector \(\overrightarrow{OP}\) is multiplied by the scale - factor \(k\).

Step2: Solve problem 1

When dilating point \(B\) with a scale factor of \(5\) and \(A\) as the center of dilation. Let \(A\) be the origin \((0,0)\) for simplicity (since the relative positions matter). If we assume \(B\) has some position vector \(\overrightarrow{AB}\), the new position vector \(\overrightarrow{AB'}=5\overrightarrow{AB}\). Looking at the relative positions of the points, the image of \(B\) is \(I\).

Step3: Solve problem 2

Let \(H\) be the center of dilation. Let the distance from \(H\) to \(C\) be \(d_{HC}\) and the distance from \(H\) to \(E\) be \(d_{HE}\). The scale factor \(k=\frac{d_{HE}}{d_{HC}}\). By observing the relative distances of the points from \(H\), the scale factor \(k = 3\).

Step4: Solve problem 3

Using \(H\) as the center of dilation, let the distance from \(H\) to \(I\) be \(d_{HI}\) and the distance from \(H\) to \(G\) be \(d_{HG}\). The scale factor \(k=\frac{d_{HG}}{d_{HI}}\). By observing the relative positions, the scale factor \(k=\frac{1}{2}\).

Step5: Solve problem 4

To dilate \(F\) so that its image is \(B\), we can use point \(A\) as the center of dilation. We can find the appropriate scale - factor by looking at the ratio of the distances from \(A\) to \(F\) and from \(A\) to \(B\).

Step6: Solve problem 5

When dilating \(H\) using \(A\) as the center of dilation and a scale factor of \(\frac{1}{3}\), we consider the vector \(\overrightarrow{AH}\) and multiply it by \(\frac{1}{3}\). The image of \(H\) is \(D\).

Step7: Solve problem 6

To take \(F\) to \(H\), we can use \(A\) as the center of dilation. The scale factor \(k\) is such that if we consider the vector \(\overrightarrow{AF}\) and \(\overrightarrow{AH}\), \(k=\frac{|\overrightarrow{AH}|}{|\overrightarrow{AF}|}\). The scale factor is \(\frac{3}{2}\), and the center of dilation is \(A\).

Step8: Solve problem 7

If we dilate \(H\) with \(B\) as the center of dilation and the image is \(H\) itself, the scale factor \(k = 1\).

Answer:

1. \(I\)
2. \(3\)
3. \(\frac{1}{2}\)
4. \(A\)
5. \(D\)
6. Center of dilation: \(A\), Scale factor: \(\frac{3}{2}\)
7. \(1\)