Transformations and Similarity Unit Test
2 of 152 of 15 Items
Question
Use the image to answer the question.
A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 3 to 12 in 1 unit increments and a y axis capped with arrows and ranging from 15 to negative 4 in 1 unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted in quadrant 1. A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 3 to 12 in 1 unit increments and a y axis capped with arrows and ranging from 15 to negative 4 in 1 unit increments. Point upper A is plotted as left parenthesis 9 comma 9 right parenthesis and point upper B is plotted as left parenthesis 6 comma 6 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends. Point upper A prime is plotted as left parenthesis 3 comma 3 right parenthesis and upper B prime is plotted as left parenthesis 2 comma 2 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends.
Determine the scale factor used in the dilation of line AB, with a center point of dilation at the origin.
(1 point)
−13
3
−3
13
The scale factor used in the dilation of line AB, with a center point of dilation at the origin, can be determined by comparing the lengths of the corresponding sides of line AB and line A'B'.
The length of line AB can be calculated using the distance formula: distance = √((x2-x1)^2 + (y2-y1)^2)
For line AB:
Point A: (9, 9)
Point B: (6, 6)
Length of AB = √((6-9)^2 + (6-9)^2)
= √((-3)^2 + (-3)^2)
= √(9 + 9)
= √18
The length of line A'B' can also be calculated using the distance formula:
For line A'B':
Point A': (3, 3)
Point B': (2, 2)
Length of A'B' = √((2-3)^2 + (2-3)^2)
= √((-1)^2 + (-1)^2)
= √(1 + 1)
= √2
The scale factor can be calculated by dividing the length of AB by the length of A'B':
Scale factor = length of AB / length of A'B'
= √18 / √2
= (√18 * √2) / (√2 * √2)
= √(18 * 2) / √(2 * 2)
= √36 / √4
= 6 / 2
= 3
Therefore, the scale factor used in the dilation of line AB, with a center point of dilation at the origin, is 3.
To determine the scale factor used in the dilation of line AB, we need to compare the lengths of AB and A' B'.
The length of AB can be calculated using the distance formula:
AB = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) = (9, 9) and (x2, y2) = (6, 6).
AB = √((6 - 9)^2 + (6 - 9)^2) = √((-3)^2 + (-3)^2) = √(9 + 9) = √18
Similarly, the length of A' B' can be calculated using the distance formula:
A' B' = √((x2' - x1')^2 + (y2' - y1')^2)
where (x1', y1') = (3, 3) and (x2', y2') = (2, 2).
A' B' = √((2 - 3)^2 + (2 - 3)^2) = √((-1)^2 + (-1)^2) = √(1 + 1) = √2
Now, we can determine the scale factor by dividing the length of AB by the length of A' B':
Scale factor = AB / A' B' = √18 / √2 = (√18 / √2) * (√2 / √2) = √(18/2) = √9 = 3
Therefore, the scale factor used in the dilation of line AB is 3.
To determine the scale factor used in the dilation of line AB with a center point of dilation at the origin, we need to compare the lengths of the original line AB and the corresponding line A'B'.
In the given information, point A is located at (9, 9) and point B is located at (6, 6). The corresponding points A' and B' are located at (3, 3) and (2, 2) respectively.
To find the length of line AB, we can use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using point A and point B, the distance of line AB is:
Distance AB = √((6 - 9)^2 + (6 - 9)^2) = √((-3)^2 + (-3)^2) = √(9 + 9) = √18
Similarly, to find the length of line A'B', we can use the distance formula with the coordinates of A' and B':
Distance A'B' = √((2 - 3)^2 + (2 - 3)^2) = √((-1)^2 + (-1)^2) = √(1 + 1) = √2
Finally, we can calculate the scale factor by dividing the length of line AB by the length of line A'B':
Scale factor = Distance AB / Distance A'B' = (√18) / (√2)
Simplifying further, the scale factor is (√18) / (√2) = (√(9 * 2)) / (√2) = (√9 * √2) / (√2) = 3
Therefore, the scale factor used in the dilation of line AB, with a center point of dilation at the origin, is 3.