line segment AB has end points A (-6,4) and B (-8,2) find the coordinates of A' and B' after a dilation with a scale factor of 1/2 centered at the orgin
To find the coordinates of A' and B' after a dilation with a scale factor of 1/2 centered at the origin, we can use the dilation formula:
A' = (1/2) * A
B' = (1/2) * B
First, let's find the coordinates of A':
A' = (1/2) * A
A' = (1/2) * (-6,4)
A' = (-3, 2)
So, the coordinates of A' are (-3, 2).
Next, let's find the coordinates of B':
B' = (1/2) * B
B' = (1/2) * (-8,2)
B' = (-4, 1)
So, the coordinates of B' are (-4, 1).
determine the scale factor used in the dilation of line segment AB, centered at the orgin
To determine the scale factor used in the dilation of line segment AB centered at the origin, we can compare the distances between the original points A and B and their images after the dilation, A' and B'.
The original line segment AB has endpoints A (-6,4) and B (-8,2). The dilated line segment has endpoints A' (-3,2) and B' (-4,1).
The distance between points A and B is calculated using the distance formula:
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
AB = √[(-8 - (-6))^2 + (2 - 4)^2]
AB = √[(-2)^2 + (-2)^2]
AB = √[4 + 4]
AB = √8
The distance between points A' and B' is calculated the same way:
A'B' = √[(-4 - (-3))^2 + (1 - 2)^2]
A'B' = √[(-1)^2 + (-1)^2]
A'B' = √[1 + 1]
A'B' = √2
The scale factor is the ratio of the distance between the original points and the distance between their images:
Scale factor = A'B' / AB
Scale factor = √2 / √8
Scale factor = √2 / (2√2)
Scale factor = 1 / 2
Therefore, the scale factor used in the dilation of line segment AB centered at the origin is 1/2.
what is the scale factor of a single dilation between the preimage ABC and image A'B'C'
To find the scale factor for the single dilation between the preimage triangle ABC and the image triangle A'B'C', we can compare the lengths of the corresponding sides of the two triangles.
Given that a triangle has three sides, we will have to calculate the scale factor for each pair of corresponding sides to determine if there is a consistent scale factor for the entire triangle.
Let's calculate the scale factor for each pair of corresponding sides:
1. AB to A'B':
AB = √[(-6 - (-7))^2 + (4 - 3)^2] = √[1^2 + 1^2] = √2
A'B' = √[(-3 - (-3))^2 + (2 - 1)^2] = √[0^2 + 1^2] = 1
Scale factor = A'B' / AB = 1 / √2 = √2 / 2
2. BC to B'C':
BC = √[(-7 - (-5))^2 + (3 - 1)^2] = √[2^2 + 2^2] = 2√2
B'C' = √[(-3 - (-1))^2 + (1 - (-1))^2] = √[2^2 + 2^2] = 2√2
Scale factor = B'C' / BC = 2√2 / 2√2 = 1
3. AC to A'C':
AC = √[(-6 - (-5))^2 + (4 - 1)^2] = √[1^2 + 3^2] = √10
A'C' = √[(-3 - (-1))^2 + (2 - (-1))^2] = √[2^2 + 3^2] = √13
Scale factor = A'C' / AC = √13 / √10 = √(13/10)
Since the scale factor for the sides AB to A'B' and BC to B'C' is 1, and the scale factor for the side AC to A'C' is not equal to 1, we can conclude that there is no uniform scale factor for the entire triangle.