Triangle ABC has vertices point A(−2,−2), 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 3
and a center point of dilation at the origin.(2 points)
To find the coordinates of A', B', and C' after a dilation with a scale factor of 3 and a center point of dilation at the origin, we need to multiply the x and y coordinates of each point by the scale factor.
For point A(-2, -2):
x-coordinate of A' = -2 * 3 = -6
y-coordinate of A' = -2 * 3 = -6
So, the coordinates of A' are (-6, -6).
For point B(-4, -3):
x-coordinate of B' = -4 * 3 = -12
y-coordinate of B' = -3 * 3 = -9
So, the coordinates of B' are (-12, -9).
For point C(-2, -5):
x-coordinate of C' = -2 * 3 = -6
y-coordinate of C' = -5 * 3 = -15
So, the coordinates of C' are (-6, -15).
Therefore, the coordinates of A', B', and C' after the dilation with a scale factor of 3 and a center point of dilation at the origin are (-6, -6), (-12, -9), and (-6, -15), respectively.
To find the coordinates of the new points after a dilation with a scale factor of 3 and a center point of dilation at the origin, we can use the formula:
A' = (k * (Ax - O), k * (Ay - O))
B' = (k * (Bx - O), k * (By - O))
C' = (k * (Cx - O), k * (Cy - O))
where A', B', and C' are the new coordinates, (Ax, Ay), (Bx, By), and (Cx, Cy) are the coordinates of the original points A, B, and C, k is the scale factor, and O is the origin (0, 0).
Given:
A(-2, -2)
B(-4, -3)
C(-2, -5)
k = 3
O(0, 0)
Let's calculate the new coordinates:
For point A:
A' = (k * (Ax - O), k * (Ay - O))
= (3 * (-2 - 0), 3 * (-2 - 0))
= (3 * (-2), 3 * (-2))
= (-6, -6)
Therefore, A' = (-6, -6).
For point B:
B' = (k * (Bx - O), k * (By - O))
= (3 * (-4 - 0), 3 * (-3 - 0))
= (3 * (-4), 3 * (-3))
= (-12, -9)
Therefore, B' = (-12, -9).
For point C:
C' = (k * (Cx - O), k * (Cy - O))
= (3 * (-2 - 0), 3 * (-5 - 0))
= (3 * (-2), 3 * (-5))
= (-6, -15)
Therefore, C' = (-6, -15).
Therefore, the coordinates of A', B', and C' after the dilation with a scale factor of 3 and a center point of dilation at the origin are:
A' = (-6, -6)
B' = (-12, -9)
C' = (-6, -15)
To find the coordinates of the vertices after a dilation, we can multiply each coordinate by the scale factor.
The scale factor is 3, so we will multiply the coordinates of each vertex by 3.
Let's find the coordinates of A':
x-coordinate of A' = x-coordinate of A * scale factor
= -2 * 3
= -6
y-coordinate of A' = y-coordinate of A * scale factor
= -2 * 3
= -6
Therefore, A' has coordinates (-6, -6).
Now, let's find the coordinates of B':
x-coordinate of B' = x-coordinate of B * scale factor
= -4 * 3
= -12
y-coordinate of B' = y-coordinate of B * scale factor
= -3 * 3
= -9
Therefore, B' has coordinates (-12, -9).
Finally, let's find the coordinates of C':
x-coordinate of C' = x-coordinate of C * scale factor
= -2 * 3
= -6
y-coordinate of C' = y-coordinate of C * scale factor
= -5 * 3
= -15
Therefore, C' has coordinates (-6, -15).
To summarize, the coordinates of A', B', and C' after a dilation with a scale factor of 3 and a center point of dilation at the origin are:
A' (-6, -6)
B' (-12, -9)
C' (-6, -15)