Please help with number 2!

Answers:

D'' = (1, -3)

E'' = (1, -1)

F'' = (3, -1)

G'' = (3, -4)

A diagram is shown below.

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Explanation:

The center of dilation is point E, which means this point does not move when the dilation is applied. Every other point will move.

As your diagram indicates, segment DE is 4 units long. If we apply the scale factor 1/2, then D'E' will be half as long meaning D'E' = 2. So point D' is 2 units above point E at (1,3)

Point F is 4 units away from point E. The scale factor 1/2 will bring F closer to E leading to segment F'E' = 2. So we'll start at E and move 2 units to the right to land on F ' (3, 1)

From F to G is 6 units, which cuts in half to 3. So from F' to G' is 3 units telling us we start at F ' (3, 1) and go up three units to get go G ' (3, 4)

So far we have

D ' = (1, 3)

E ' = (1, 1) ... fixed point doesn't move

F ' = (3, 1)

G ' = (3, 4)

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From here we'll apply the reflection over the x axis rule which says

the x coordinate stays the same, and the y coordinate flips from positive to negative (or vice versa).

Based on what was mentioned for D', E', F' and G', we get the following

D'' = (1, -3)

E'' = (1, -1) .... this point does move now

F'' = (3, -1)

G'' = (3, -4)

Points E' and E'' are in different locations because all fixed points in this reflection are along the mirror line y = 0 (aka the x axis). In other words, if E was on the x axis, then it would not move if we applied this reflection.

Check out the diagram below see a visual summary of what happened. Note how the red points moved in closer to point E (since the distances from the center E have been cut in half) compared to the blue counterparts.