Question 1034516: A square has 12cm sides. Describe its imag for a dilation with center at one of the vertices and scale factor of 0.4.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if the center is at one of the vertices, then the other vertices on a graph of the square would be:
(0,0)
(0,12)
(12,12)
(12,0)
the dilation of .4 would be multiplying all the vertices by .4
.4 * 12 = 4.8.
the new coordinates would be:
(0,0)
(0,4.8)
(4.8,4.8)
(4.8,0)
the dilation is done by distance from the point of reference.
since the point of reference is (0,0), then the dilation is done by distance from that point of reference.
the distance on the graph is d = sqrt((x2-x1)^2 + (y2-y1)^2).
when x1 and y1 = 0, as in the coordinate (0,0), the formula becomes:
d = sqrt(x^2 + y^2).
when x = 0 and y = 12, the formula becomes d = sqrt(0^2 + 12^2) = sqrt(12^2) = 12.
when you dilate it, you are multiplying the distance by the scale factor which is .4
.4 * 12 = 4.8.
the most difficult calculation comes in when x = 12 and y = 12.
d = sqrt(12^2 + 12^2)
when you multiply d by .4, you get the new d = .4 * sqrt(12^2 + 12^2).
if you square d, you get d^2 = .4^2 * sqrt(12^2 + 12^2)^2.
you can simplify this to get d^2 = .16 * (12^2 + 12^2).
simplify this further to get d^2 = .16 * 288 = 46.08
this is the hypotenuse of a right triangle where d is the hypotenuse and x is the adjacent side and y is the opposite side of the angle formed from the center of the graph.
the right triangle formula for the hypotenuse is d^2 = x^2 + y^2.
since d^2 = 46.08, and since x = y, the formula becomes 46.08 = 2x^2
solve for x^2 to get x^2 = 23.04.
solve for x to get x = sqrt(23.04) = 4.8.
this means that y = 4.8 as well.
the coordinate of (12,12) becomes (4.8,4.8).
if you dilate from the origin, then all dilations are from that point as the reference.
if you dilate from any other point, than all dilations are from that other point as the reference.
dilations from the origin are the easiest to perform, but the process would be the same from any other point.
in that case, the distance formula has to be sqrt((x2-x1)^2 + (y2-y1)^2).
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