What fraction of the total # of students earned at least a C? Include small explanation on how you did it.
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The dilation by a scale factor of 2 of the points A(2, 3), B(5, 4), C(3, 6) gives;
a. A'(4, 6), B'(10, 8), C'(6, 12)
b. A'(-2, -2), B'(4, 0), C'(0, 4)
c. The transformation that would carry dilation 1 onto dilation 2 is T(-6, -8)
- The area of dilation 1 and 2 are the same
- The center of dilation does not change the area
d. The proportion of the side length of Dilation 1 and Dilation 2 is 1:1
- The angle measures are the same
The general formula for finding the coordinates of the image of a point following a dilation is presented as follows;
Where;
(a, b) = The center of dilation
k = The scale factor of dilation
(x, y) = The coordinate of the pre-image
The given points are;
A(2, 3), B(5, 4), C(3, 6)
a. The scale factor of dilation = 2
The center of dilation = The origin (0, 0)
Therefore;
Therefore dilation about the origin, with a scale factor of 2 gives;
- A(2, 3) → A'(4, 6)
Similarly
- B(5, 4) → B'(10, 8)
- C(3, 6) → C'(6, 12)
b. With the center of dilation at (6, 8), we have;
- A(2, 3) → A'(-2, -2)
- B(5, 4) → B'(4, 0)
- C(3, 6) → C'(0, 4)
c. The difference between the coordinates of the points on dilation 1 and 2 is a shift left 6 places and a shift downwards 8 places
Using notation, we have;
- Dilation 1 T(-6, -8) → Dilation 2
The area of the images of dilation 1 and 2 are equal given that the scale factor is the same.
- The location of the center of dilation does not change the area of the image
d. From the above calculation, given that the difference between pre-image point and the center is multiplied by the scale factor followed by the addition of the <em>x </em>and y-values, the lengths of the sides of dilation 1 and 2 are the same, such that we have;
- The proportion of the side lengths is 1
Given that the side lengths are the same, by AAA congruency postulate, we have;
- The angle measures are the same.
Learn more about dilation transformation here:
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