Math  /  Geometry

Question*\#5.) a.) If ABCDA B C D undergoes R180R_{-180^{\circ}} about the origin, what would be the coordiante point of AA^{\prime} ? a.) (5,5)(-5,5) b.) (3,5)(-3,-5) c.) (5,3)(-5,3) d.) (3,3)(-3,3) b.) Graph the image EFGHE F G H after R270R_{270}.

Studdy Solution

STEP 1

What is this asking? Where does point A end up after spinning the square ABCD half a turn around the origin, and what does the square EFGH look like after spinning it three-quarters of a turn counter-clockwise? Watch out! Don't mix up clockwise and counter-clockwise rotations!
Also, double-check your signs when rotating points.

STEP 2

1. Rotate Point A
2. Rotate Square EFGH

STEP 3

Alright, let's **locate point A** on the graph!
It's at (3,5)(3, 5), right?
So, its x-coordinate is 33 and its y-coordinate is 55.

STEP 4

Now, we're doing a R180R_{-180^{\circ}} rotation, which means we're spinning it **halfway** around but *clockwise*!
The rule for a 180180^{\circ} rotation, whether clockwise or counter-clockwise, is to flip the signs of *both* the x and y coordinates.

STEP 5

So, if A is at (3,5)(3, 5), then A' (A prime, that's how we write the new point after rotation) will be at (3,5)(-3, -5).
Boom! We flipped the signs!

STEP 6

Okay, time to spin EFGH!
This time it's R270R_{270}, which means a **270-degree counter-clockwise** rotation.
That's like three-quarters of a full turn to the *left*!

STEP 7

Let's imagine where each point of EFGH goes.
A helpful trick for a 270270^{\circ} counter-clockwise rotation is to swap the x and y coordinates, and then change the sign of the *new* x-coordinate.

STEP 8

Since the problem doesn't give us the coordinates of EFGH, we can just describe the rotation.
Imagine turning the square three-quarters of a turn to the left.
That's where E'F'G'H' will end up!
Draw it out, and you'll see it!

STEP 9

The coordinates of A' after a R180R_{-180^{\circ}} rotation are (3,5)(-3, -5).
The image E'F'G'H' is the square EFGH rotated 270270^{\circ} counter-clockwise.
You'll need to draw that one out yourself!

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