Question

1. draw the images of points p and q using c as the center of dilation and a scale factor of 4. label the new points p and q.
2. draw the images of points p and q using c as the center of dilation and a scale factor of 1/4. label the new points p and q.

reflection questions:
what are the effects of using a scale factor greater than one?
what are the effects of using a scale factor of less than one?
how would you explain the steps for dilating a point?
cool down: a single dilation of a triangle
desales drew a triangle and a dilation of the triangle with scale factor 1/2:

1. what is the center of the dilation? explain how you know.
2. which triangle is the original and which triangle is the dilation? explain how you know.

Explanation:

1. Center of Dilation:

• In a dilation, the center of dilation is the fixed point from which distances are measured. For two similar figures (in this case, triangles) formed by dilation, the lines connecting corresponding vertices pass through the center of dilation.
• Looking at the triangles \( \triangle ABC\) and \( \triangle ADE\), if we extend the lines \(CD\) and \(BE\) (or consider the fact that the ratio of side - lengths is consistent with dilation), we can observe that the lines connecting corresponding vertices (e.g., if we assume \(A\) is a common - like point in the sense of dilation mapping, but more accurately, for a non - degenerate dilation of triangles, the intersection of lines joining corresponding vertices gives the center. Here, if we consider the property of dilation, the center of dilation is \(A\). Because when we perform a dilation, all points of the original figure and their images lie on lines passing through the center of dilation. For example, if we consider the sides \(AB\) and \(AE\) (assuming a proper correspondence based on scale factor \(\frac{1}{2}\)), the line \(BE\) and other lines joining corresponding points (if we assume \(C\) corresponds to \(D\) in the dilation with scale factor \(\frac{1}{2}\)) will pass through \(A\).

1. Original and Dilated Triangle:

• The scale factor of dilation is \(k = \frac{1}{2}\). When \(0\lt k\lt1\), the dilation is a reduction.
• The larger triangle is the original triangle. Since the scale factor \(\frac{1}{2}\) is less than \(1\), the triangle that is larger in size is the original one. So, \( \triangle ABC\) is the original triangle and \( \triangle ADE\) is the dilated triangle. Mathematically, if we assume the side - length relationship \(AD=\frac{1}{2}AC\) (based on the scale factor of dilation), the triangle \( \triangle ABC\) has longer side - lengths (e.g., \(AC\gt AD\) if the dilation is from \( \triangle ABC\) to \( \triangle ADE\) with scale factor \(\frac{1}{2}\)).

Answer:

1. The center of the dilation is \(A\). We know this because in a dilation, lines connecting corresponding vertices of the original and dilated figures pass through the center of dilation. For the triangles in the figure, the lines (implied by the side - length relationships and the property of dilation) pass through \(A\).
2. \( \triangle ABC\) is the original triangle and \( \triangle ADE\) is the dilated triangle. Since the scale factor of dilation \(k = \frac{1}{2}\lt1\), the dilation is a reduction. The larger triangle (\( \triangle ABC\)) is the original, and the smaller triangle (\( \triangle ADE\)) is the result of the dilation.