Question
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)Responses22−2negative 212Start Fraction 1 over 2 End Fraction−12
Answers
Answered by
GPT 3.5
The scale factor for the dilation can be found by comparing the corresponding side lengths of the original triangle ABC and the dilated triangle A'B'C'.
Let's first calculate the side lengths of triangle ABC:
AB = sqrt((-6 - (-2))^2 + (8 - 4)^2) = sqrt(16 + 16) = sqrt(32)
BC = sqrt((-2 - (-6))^2 + (4 - 8)^2) = sqrt(16 + 16) = sqrt(32)
AC = sqrt((-2 - (-2))^2 + (4 - 4)^2) = sqrt(0 + 0) = 0 (note: length of AC is 0 as points A and C coincide)
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)
B'C' = sqrt((-1 - (-3))^2 + (2 - 4)^2) = sqrt(4 + 4) = sqrt(8)
A'C' = sqrt((-1 - (-1))^2 + (2 - 2)^2) = sqrt(0 + 0) = 0 (note: length of A'C' is 0 as points A' and C' coincide)
The scale factor can be determined as the ratio of the corresponding side lengths:
Scale factor = A'B'/AB = sqrt(8)/sqrt(32) = sqrt(8/32) = sqrt(1/4) = 1/2
Therefore, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin is 1/2.
Let's first calculate the side lengths of triangle ABC:
AB = sqrt((-6 - (-2))^2 + (8 - 4)^2) = sqrt(16 + 16) = sqrt(32)
BC = sqrt((-2 - (-6))^2 + (4 - 8)^2) = sqrt(16 + 16) = sqrt(32)
AC = sqrt((-2 - (-2))^2 + (4 - 4)^2) = sqrt(0 + 0) = 0 (note: length of AC is 0 as points A and C coincide)
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)
B'C' = sqrt((-1 - (-3))^2 + (2 - 4)^2) = sqrt(4 + 4) = sqrt(8)
A'C' = sqrt((-1 - (-1))^2 + (2 - 2)^2) = sqrt(0 + 0) = 0 (note: length of A'C' is 0 as points A' and C' coincide)
The scale factor can be determined as the ratio of the corresponding side lengths:
Scale factor = A'B'/AB = sqrt(8)/sqrt(32) = sqrt(8/32) = sqrt(1/4) = 1/2
Therefore, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin is 1/2.