Question
¯¯¯¯LM¯¯¯¯¯ is dilated by a scale factor of 4, with a center of dilation at L. If L is at point (2,4) and M is at point (5,3) , where would L′ be located?(1 point) Responses (−0.5, 1) open paren negative 0 point 5 comma 1 close paren (20,12) open paren 20 comma 12 close paren (8, 16) open paren 8 comma 16 close paren (2,4)
Answers
GPT-4o mini
When a point is dilated from a center of dilation, the new coordinates of the point can be found by using the formula:
\[
(x', y') = (x_c + k(x - x_c), y_c + k(y - y_c))
\]
where \((x_c, y_c)\) is the center of dilation, \(k\) is the scale factor, and \((x, y)\) are the original coordinates of the point being dilated.
In this case:
- The center of dilation \(L\) is at \((2, 4)\).
- The scale factor \(k\) is \(4\).
- The coordinates of point \(L\) that we are dilating are \((2, 4)\).
Plugging these values into the formula:
\[
(x', y') = (2 + 4(2 - 2), 4 + 4(4 - 4))
\]
This simplifies to:
\[
(x', y') = (2 + 4(0), 4 + 4(0)) = (2 + 0, 4 + 0) = (2, 4)
\]
Therefore, the location of \(L'\) after dilation is \((2, 4)\).
Thus, the answer is: (2, 4).
\[
(x', y') = (x_c + k(x - x_c), y_c + k(y - y_c))
\]
where \((x_c, y_c)\) is the center of dilation, \(k\) is the scale factor, and \((x, y)\) are the original coordinates of the point being dilated.
In this case:
- The center of dilation \(L\) is at \((2, 4)\).
- The scale factor \(k\) is \(4\).
- The coordinates of point \(L\) that we are dilating are \((2, 4)\).
Plugging these values into the formula:
\[
(x', y') = (2 + 4(2 - 2), 4 + 4(4 - 4))
\]
This simplifies to:
\[
(x', y') = (2 + 4(0), 4 + 4(0)) = (2 + 0, 4 + 0) = (2, 4)
\]
Therefore, the location of \(L'\) after dilation is \((2, 4)\).
Thus, the answer is: (2, 4).