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In the diagram, polygon ABCD is flipped over a line of reflection to make a polygon with vertices at A′, B′, C′, and D′. Points A', B', and D' are shown, but the line of reflection and point C′ are not. What are the coordinates of C'′ ?

A(6, 2)

B(5, 2)

C(7, 2)

D.(12, 2)

In the diagram, polygon ABCD is flipped over a line of reflection to make a polygon with vertices at A , B , C , and D . Points A , B , and D are shown, but the class=

Verified Answer :

when flipped over a line of reflection the lengths are still the same
the point to the line of reflection is the same length as the line of reflection to the reflected position
the distance from the original point to the reflected point is twice the distance from the original point to the line of reflection. 

The correct answer is:


A) (6, 2)


Explanation:


The coordinates of ABCD are as follows:

A(1, 6)

B(1, 2)

C(4, 2)

D(3, 5)


The coordinates of A'B'C'D' are as follows:

A'(9, 6)

B'(9, 2)

C' -- ??

D'(7, 5)


Comparing the pre-image points to the image points, we see that the y-coordinates do not change; only the x-coordinates change. This means we will be reflecting across a vertical line.


Vertical lines are of the form x=c, where c is some constant. To find this value, we average the x-coordinates of each point:


For A: (1+9)/2 = 10/2 = 5

For B: (1+9)/2 = 10/2 = 5

For D: (3+7)/2 = 10/2 = 5


Each time, the average is 5; this means the line x=5 is directly between the polygons, and is our line of reflection.


We know the y-coordinate of C' will be the same as C, which is 2. This gives us (x, 2) for the ordered pair.


Using the line of reflection, we will set up an equation to find the x-coordinate of C':

(4+x)/2 = 5


First, multiply each side by 2:

((4+x)/2)2 = 52

4+x = 10


Subtract 4 from each side:

4+x-4 = 10-4

x = 6


The coordinates of C' are (6, 2).

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