Triangle ABC has vertices point A(2,1) , point B(4,3) , and point C(−2,5) . Find the coordinates of A′ , B′ , and C′ after a dilation with a scale factor of 7 and a center point of dilation at the origin (0,0) .(2 points)
To find the coordinates of A', B', and C' after a dilation with a scale factor of 7 and a center point of dilation at the origin (0,0), we can use the following formula:
A' = (7 * Ax, 7 * Ay)
B' = (7 * Bx, 7 * By)
C' = (7 * Cx, 7 * Cy)
Given the coordinates of the vertices:
A(2,1), B(4,3), and C(-2,5)
1. For A':
A' = (7 * 2, 7 * 1) = (14, 7)
2. For B':
B' = (7 * 4, 7 * 3) = (28, 21)
3. For C':
C' = (7 * -2, 7 * 5) = (-14, 35)
So the coordinates of A', B', and C' after the dilation are:
A' = (14, 7)
B' = (28, 21)
C' = (-14, 35)
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)
To determine the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0,0), we can use the formula:
Scale factor = Length of corresponding side in A'B'C' / Length of corresponding side in ABC.
From the given coordinates:
A(2, 1), B(-2, 2), and C(-3, -1)
and the coordinates after dilation:
A'(4, 2), B'(-4, 4), and C'(-6, -2)
We can calculate the lengths of the corresponding sides:
Length of AB = sqrt((2 - (-2))^2 + (1 - 2)^2) = sqrt(16 + 1) = sqrt(17)
Length of A'B' = sqrt((4 - (-4))^2 + (2 - 4)^2) = sqrt(64 + 4) = sqrt(68)
Length of BC = sqrt((-2 - (-3))^2 + (2 - (-1))^2) = sqrt(1 + 9) = sqrt(10)
Length of B'C' = sqrt((-4 - (-6))^2 + (4 - (-2))^2) = sqrt(4 + 36) = sqrt(40)
Length of AC = sqrt((2 - (-3))^2 + (1 - (-1))^2) = sqrt(25 + 4) = sqrt(29)
Length of A'C' = sqrt((4 - (-6))^2 + (2 - (-2))^2) = sqrt(100 + 16) = sqrt(116)
Now, we can find the scale factor:
Scale factor = Length of A'B' / Length of AB
= sqrt(68) / sqrt(17)
= sqrt(68/17)
= sqrt(4)
= 2
Therefore, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0,0) is 2.