Triangle ABC has vertices point A(2, 1), point B(−2, 2), and point C(−3,−1). After dilation, triangle A′B′C′ has vertices point A′(4, 2), point B′(−4, 4), and point C′(−6,−2). Determine the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0, 0 ).(1 point) The scale factor is .
To find the scale factor for the dilation, we can compare the distances between the corresponding points before and after the dilation.
The distance between points A(2, 1) and A'(4, 2) is sqrt((4-2)^2 + (2-1)^2) = sqrt(2^2 + 1^2) = sqrt(5).
The distance between points B(-2, 2) and B'(-4, 4) is sqrt((-4-(-2))^2 + (4-2)^2) = sqrt((-2)^2 + 2^2) = sqrt(8).
The distance between points C(-3, -1) and C'(-6, -2) is sqrt((-6-(-3))^2 + (-2-(-1))^2) = sqrt((-3)^2 + 1^2) = sqrt(10).
Since the origin is the center of dilation, the scale factor is the same for all distances. So, we can choose any of the distances calculated above.
Let's choose the distance between points A and A', which is sqrt(5). Therefore, the scale factor for the dilation is sqrt(5).