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Answer:
- scale factor: 3
- rule: (x, y) ⇒ (3x +15, 3y -24)
- center: (-7.5, 12)
Explanation:
The scale factor can be found by comparing the length of CB to the length of RQ.
B-C = (-2, 8) -(-4, 9) = (2, -1)
Q-R = (9, 0) -(3, 3) = (6, -3)
The length of RQ is clearly 3 times the length of CB, so the scale factor (k) is ...
ratio of corresponding differences = 6/2 = -3/-1 = 3 . . . . . scale factor
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We know that for dilation about a point O, the distance from O is multiplied by the scale factor. For dilation of point B to point Q, this means ...
k(B -O) = (Q -O)
Solving for Q, we get ...
Q = kB -kO +O . . . . this is what our dilation rule will look like.
The quantity O-kO can be found by subtracting kB:
Q -kB = O -kO = O(1 -k)
This is what we need for our dilation rule.
Q -kB = (9, 0) -3(-2, 8) = (9+6, 0-24) = (15, -24)
So, our dilation rule is ...
(x, y) ⇒ k(x, y) +(15, -24)
(x, y) ⇒ (3x +15, 3y -24) . . . . . dilation rule
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The center of dilation can be found from ...
(Q -kB)/(1 -k) = O
O = (15, -24)/(1 -3) = (-7.5, 12)
The center of dilation is (-7.5, 12).
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Additional comments
On the graph, the center of dilation can be found by drawing a line through a point and its image. (Points are always dilated along a line through the center of dilation.) The intersection of two such lines is the center of dilation.
On this graph, the center is just above the top edge of the chart, at point (-7.5, 12). You can see this if you carefully draw lines BQ and AP.
You usually have coordinates for two original points and two image points, so finding the scale factor the way we did is not difficult. If you just have one point on the original and the image, the scale factor is found by finding their distances from the center of dilation. Each image point is k times as far as each original point from that center.