Question

What set of transformations could be applied to rectangle ABCD to create A'B'C'D'?

Answer 1

@Hero

Answer 2

I don't think its any of these actually if we interpret it literally.

Answer 3

well what do you think is the best option

Answer 4

In order for it to get from the 2nd Quad to the 4th Quad, there would need to be a reflection, a rotation AND a shift.

Answer 5

I agree with @Hero. None of them really map the rectangle completely.

Answer 6

The best is probably reflected over the x-axis and then rotated.

Answer 7

rotated 90 or 180? probably 90 huh

Answer 8

Definitely not 180.

Answer 9

If rectangle ABCD was reflected over the y-axis, reflected over the x-axis, and rotated 180°, where would point A' lie?

Answer 10

@Hero

Answer 11

Let's take it one step at a time. Where would point A be after reflection about the y-axis?

Answer 12

4,1

Answer 13

Correct. Where would point A be after reflection over the x-axis?

Answer 14

4,-1?

Answer 15

Correct. Now ...

Answer 16

nows the hard part for me lol

Answer 17

This is the part that is tricky for you.

Answer 18

So here's the trick for this ... I'm going to try to explain. It's really easy once you know the trick.

Answer 19

Geometrically, what does 180 degrees represent?

Answer 20

what do you mean

Answer 21

half?

Answer 22

It represents a straight line.

Answer 23

A straight line the goes through the origin.

Answer 24

oh okay

Answer 25

They don't tell you that but whenever they say "180 degree rotation" they are implying that the rotation will occur about the origin.

Answer 26

And so there is an imaginary line where A and A' will both be on that same line.

Answer 27

Also, the distance of the line segment from A to O (the origin) will be the same as that from O to A'

Answer 28

would it be (-4,1)?

Answer 29

Correct!

Answer 30

Yay!

Answer 31

So in other words, A and A' are the same point.

Answer 32

theres only one more i don't get after this!

Answer 33

Also, we should re-visit that first question after this.

Answer 34

sounds good

Answer 35

First I should ask, which options do you think we might be able to eliminate as definitely not correct?

Answer 36

A

Answer 37

B probably as well

Answer 38

Why would you eliminate those choices?

Answer 39

because of how large the rotations are i think it would only be 90

Answer 40

They probably didn't tell you this before, but when you reflect an object over the x-axis, then reflect it over the y-axis, that is the equivalent of rotating that same object 180 degrees about the origin.

Answer 41

Okay

Answer 42

When given two objects ABCD and A'B'C'D and a set of questions involving rotations, I usually look for that scenario first. I'd look to see if the objects are rotated 180 degrees about the origin. To do that, you can test each point on ABCD.

Answer 43

No they aren't because (x,y) doesn't equal (-x,-y)

Answer 44

Or rather I'd test to see if the objects are "opposites" of each other.

Answer 45

Are you sure about that? Let's take a look at point A. What are the coordinates for A?

Answer 46

haha didn't even notice that but (-4,1)

Answer 47

the other is 4, -1

Answer 48

so it would be a?

Answer 49

so A = (x,y) = (-4,1) and (-x,-y) = (-(-4),-(1)) = (4,-1) = A'. But anyway (x,y) won't equal (-x,-y) in this case since A and A' are not in the same quadrant.

Answer 50

But we do see that A and A' are opposites of each other.

Answer 51

If you test the rest of the points you'd get a similar result. Nevertheless, you are right. A is the correct choice.

Answer 52

Okay cool, so now back to the first question?

Answer 53

Okay, so for the first question, what can answer choices can you eliminate?

Answer 54

i don't really know

Answer 55

B

Answer 56

@Hero

Answer 57

Sorry, I was afk for a bit.

Answer 58

all good

Answer 59

Why did you eliminate B?

Answer 60

because the coordinates don't match the 180 degree rotation rule

Answer 61

Very good. What else can you eliminate?

Answer 62

A

Answer 63

wait no

Answer 64

is the answer a lol

Answer 65

Actually, A can also be eliminated because when you reflect the object over the x-axis, then reflect it over the origin, it would end up in the 1st quadrant, but A'B'C'D' is not in the first quadrant.

Answer 66

even though the coordinates go from (-1,-1) to (1,1) ??

Answer 67

for point C

Answer 68

You're only looking at one point.

Answer 69

oh never mind thats one point

Answer 70

You have to check ALL the points.

Answer 71

yeah

Answer 72

okay so c and d

Answer 73

It's between C and D at this point. Any ideas on it?

Answer 74

i really have no idea

Answer 75

So when they talk about rotating an object 90 degrees, what the mean is 90 degrees with respect to the origin. In other words, geometrically, if the origin is presented by the point O, then each point A to A', B to B', C to C' and D to D' will be at right angles to each other after the initial reflection about the given axis is made.

Answer 76

wouldn't it be C?

Answer 77

So when solving problems like these, first we try to eliminate two of the options. The next step is to "reason" between the last two options, and then we will decide which option (A, B, C or D) is the correct option. For now, we're still at the "reasoning" step.

Answer 78

But we can "consider option C" by exploring the particular scenario that is presented with this option.

Answer 79

So considering option C, does it fit want we want as the correct outcome? Why or why not? This is what we want to analyze.

Answer 80

because after reflecting over the x axis and d doing a 90 degree rotation counterclockwise that is where it would be right?

Answer 81

Basically, after reflecting the object over the x-axis, if that option is correct, you should be able to draw a right angle from point A to A', point B to B', etc.

Answer 82

with the right angle itself occurring at the origin.

Answer 83

i still think c

Answer 84

We have to have a reason to eliminate D.

Answer 85

Ok well i don't know what that reason is

Answer 86

wait no

Answer 87

after checking the points on it, think it follows the rules of a 90 rotation best for D

Answer 88

it matches at ever point for (x,y) > (y,-x)

Answer 89

But, is the rotation for D clockwise or counterclockwise?

Answer 90

ohhh

Answer 91

so then it follows (-y,x) for c

Answer 92

You didn't answer my question. Is D a clockwise rotation or a counterclockwise rotation?

Answer 93

counter clockwise