What is the image of the point A(2, 3) under the dilation by factor 3 with center of dilation at the point (1, -2)?
Question
What is the image of the point A(2, 3) under the dilation by factor 3 with center of dilation at the point (1, -2)?
Solution
To find the image of a point under dilation, we use the formula:
A'(x', y') = O + k(A - O)
where:
- A'(x', y') is the image of the point A under dilation,
- O is the center of dilation,
- k is the dilation factor,
- A is the original point.
Here, A(2, 3), O(1, -2), and k = 3.
First, we find the vectors OA and O'A':
OA = A - O = (2, 3) - (1, -2) = (2-1, 3-(-2)) = (1, 5)
O'A' = k * OA = 3 * (1, 5) = (3, 15)
Then, we find the coordinates of A' by adding the vector O'A' to the coordinates of O:
A' = O + O'A' = (1, -2) + (3, 15) = (1+3, -2+15) = (4, 13)
So, the image of the point A(2, 3) under the dilation by factor 3 with center of dilation at the point (1, -2) is A'(4, 13).
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