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, we can compare the corresponding side lengths of the original triangle ABC and the dilated triangle A'B'C'.
The distance formula can be used to find the length of a side. For example, the length of side AB is:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-2 - 2)^2 + (2 - 1)^2)
= sqrt(16 + 1)
= sqrt(17)
Similarly, we can find the lengths of sides BC and AC.
For the dilated triangle, the length of side A'B' is:
A'B' = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-4 - 4)^2 + (4 - 2)^2)
= sqrt(64 + 4)
= sqrt(68)
We can also find the lengths of sides B'C' and A'C'.
Now, let's compare the corresponding side lengths:
AB / A'B' = sqrt(17) / sqrt(68)
BC / B'C' = ...
AC / A'C' = ...
Since the center point of dilation is the origin (0, 0), the scale factor is determined by the length ratios.