Triangle ABC has vertices point A (−2,4) , point B (−6,8) , and point C (−2,4) . After dilation, triangle A′B′C′ has vertices point A′ (−1,2) , point B′ (−3,4) , and point C′ (−1,2) . Determine the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0,0) . (1 point)

Question

Responses

−2
negative 2

2
2

−12
negative Start Fraction 1 over 2 End Fraction

12

Answer 1

To find the scale factor of the dilation, we can calculate the ratio of the corresponding side lengths of the original triangle ABC and the dilated triangle A'B'C'.

Let's calculate the side lengths of triangle ABC first:
AB = sqrt((-6 - (-2))^2 + (8 - 4)^2) = sqrt(16 + 16) = sqrt(32) = 4sqrt(2)
BC = sqrt((-2 - (-6))^2 + (4 - 8)^2) = sqrt(16 + 16) = sqrt(32) = 4sqrt(2)
AC = sqrt((-2 - (-2))^2 + (4 - 4)^2) = sqrt(0 + 0) = 0

Now let's calculate the side lengths of triangle A'B'C':
A'B' = sqrt((-3 - (-1))^2 + (4 - 2)^2) = sqrt(4 + 4) = sqrt(8) = 2sqrt(2)
B'C' = sqrt((-1 - (-1))^2 + (2 - 2)^2) = sqrt(0 + 0) = 0
A'C' = sqrt((-3 - (-1))^2 + (4 - 2)^2) = sqrt(4 + 4) = sqrt(8) = 2sqrt(2)

Now we can calculate the scale factor:
scale factor = A'B' / AB = (2sqrt(2)) / (4sqrt(2)) = 2sqrt(2) / (2sqrt(2) * sqrt(2)) = 1 / sqrt(2) = sqrt(2) / 2

Therefore, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin is sqrt(2) / 2 or 1 / (2sqrt(2)).