A quadrilateral has the following coordinates.
E (4,7) ; F (7,5) ; G (8,14) ; H (5,9)
Quadrilateral EFGH is transformed and the coordinates of the new figure are shown below.
E' (12,21); F' (21, 15) ;G' (24,42); H' (15,27)
Is the transformation a dilation?
A.
No, because point H'(15 , 27) is incorrect.
B.
No, because point G'(24 , 42) is incorrect.
C.
Yes, because all points were multiplied by a scale factor of 8.
D.
Yes, because all points were multiplied by a scale factor of 3.
C. Yes, because all points were multiplied by a scale factor of 8.
D.
Yes, because all points were multiplied by a scale factor of 3.
To determine if the transformation is a dilation, we need to consider two factors: whether the points were uniformly scaled and whether the scale factor is consistent for all points.
First, let's calculate the scale factor by comparing the corresponding coordinates in the original and transformed figures. We can choose any pair of points, but let's take the points E and E'.
To find the x-scale factor, we divide the x-coordinate of E' by the x-coordinate of E:
x-scale factor = x-coordinate of E' / x-coordinate of E
= 12 / 4
= 3
To find the y-scale factor, we divide the y-coordinate of E' by the y-coordinate of E:
y-scale factor = y-coordinate of E' / y-coordinate of E
= 21 / 7
= 3
Since both the x-scale factor and y-scale factor are equal to 3, we can conclude that the scale factor is consistent for all points.
Now, let's examine the answer choices:
A. No, because point H'(15, 27) is incorrect.
Since the point H' matches the transformation, this answer is incorrect.
B. No, because point G'(24, 42) is incorrect.
Similarly, since the point G' matches the transformation, this answer is also incorrect.
C. Yes, because all points were multiplied by a scale factor of 8.
The correct scale factor based on our calculation is 3, not 8. Therefore, this answer is incorrect.
D. Yes, because all points were multiplied by a scale factor of 3.
Based on our calculation of the scale factor, this answer is correct. The transformation is a dilation with a scale factor of 3.
Therefore, the correct answer is D.