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 need to find the ratio of the corresponding side lengths of the two triangles.
The distance formula can be used to find the side lengths of triangle ABC:
AB = sqrt((-2-2)^2 + (2-1)^2) = sqrt(16+1) = sqrt(17)
BC = sqrt((-3+2)^2 + (-1-2)^2) = sqrt(1+9) = sqrt(10)
AC = sqrt((-3-2)^2 + (-1-1)^2) = sqrt(25+4) = sqrt(29)
Similarly, we can find the side lengths of triangle A'B'C':
A'B' = sqrt((-4-4)^2 + (4-2)^2) = sqrt(64+4) = sqrt(68)
B'C' = sqrt((-6+4)^2 + (-2-4)^2) = sqrt(4+36) = sqrt(40)
A'C' = sqrt((-6-4)^2 + (-2-2)^2) = sqrt(100+16) = sqrt(116)
Now we can find the ratio of the corresponding side lengths:
AB/A'B' = sqrt(17)/sqrt(68) = 1/2
BC/B'C' = sqrt(10)/sqrt(40) = 1/2
AC/A'C' = sqrt(29)/sqrt(116) = 1/2
Since all three ratios are equal to 1/2, the scale factor for the dilation is 1/2.