Triangle ABC has verticals point A (2,1), B (-2,2), and C (-3,-1). After dilation, triangle A’B’C’ has a verticals 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). The scale factor is
To determine the scale factor of a dilation, we can compare the corresponding side lengths of the original triangle ABC and the dilated triangle A'B'C'.
Let's start with side AB. The length of side AB is given by the distance formula:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-2 - 2)^2 + (2 - 1)^2)
= sqrt(16 + 1)
= sqrt(17)
Similarly, the length of side A'B' is given by:
A'B' = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-4 - 4)^2 + (4 - 2)^2)
= sqrt(64 + 4)
= sqrt(68)
= 2sqrt(17)
The scale factor can be calculated by dividing the length of A'B' by the length of AB:
Scale factor = A'B' / AB
= 2sqrt(17) / sqrt(17)
= 2
Therefore, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin is 2.