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)
To find the scale factor for the dilation of triangle ABC, we can compare the coordinates of corresponding vertices in triangle ABC and triangle A'B'C'.
The change in x-coordinate for point A is 4 - 2 = 2. The change in y-coordinate for point A is 2 - 1 = 1. Therefore, the scale factor for the dilation of triangle ABC with respect to point A is 2:1.
The change in x-coordinate for point B is -4 - (-2) = -2. The change in y-coordinate for point B is 4 - 2 = 2. Therefore, the scale factor for the dilation of triangle ABC with respect to point B is also 2:1.
The change in x-coordinate for point C is -6 - (-3) = -3. The change in y-coordinate for point C is -2 - (-1) = -1. Therefore, the scale factor for the dilation of triangle ABC with respect to point C is also 3:1.
Since all three scale factors (2:1, 2:1, and 3:1) are the same, we can conclude that the overall scale factor for the dilation of triangle ABC with a center point of dilation at the origin is 2:1.
wrong bot
Apologies for the mistake. Let's find the scale factor correctly.
To find the scale factor, we can compare the lengths of corresponding sides in triangle ABC and A'B'C'.
Let's compare the lengths of side AB. The length of AB is determined by the distance formula:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For triangle ABC: AB = sqrt((-2-2)^2 + (2-1)^2) = sqrt(16+1) = sqrt(17)
For triangle A'B'C': A'B' = sqrt((-4-4)^2 + (4-2)^2) = sqrt(64+4) = sqrt(68)
The scale factor for side AB is given by: A'B' / AB = sqrt(68) / sqrt(17)
Similarly, we can find the scale factors for sides BC and AC and verify if they are the same.
BC scale factor: B'C' / BC = sqrt((-4-(-3))^2 + (4-(-1))^2) / sqrt((-2-2)^2 + (2-1)^2)
= sqrt(17) / sqrt(17) = 1
AC scale factor: A'C' / AC = sqrt((-6-2)^2 + (-2-1)^2) / sqrt((-2-2)^2 + (2-1)^2)
= sqrt(45) / sqrt(17)
Since the scale factors for the three sides are not all the same, we cannot determine a single scale factor for the dilation of triangle ABC with a center point of dilation at the origin.