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Let's solve each problem step by step and then answer the final question about multiple transformations.

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Problem 1: Translate △QRS by the rule (x, y) → (x – 3, y – 4)

Given:
- Q(4,1)
- R(1,-2)
- S(2,3)

Apply the translation: subtract 3 from x and 4 from y.

- Q': (4 – 3, 1 – 4) = (1, -3)
- R': (1 – 3, -2 – 4) = (-2, -6)
- S': (2 – 3, 3 – 4) = (-1, -1)

✔ So:
- Q'(1, -3)
- R'(-2, -6)
- S'(-1, -1)

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Problem 2: Reflect △Q'R'S' over the x-axis

We now reflect the points from Problem 1 over the x-axis.
Reflection over x-axis: (x, y) → (x, -y)

From Problem 1:
- Q'(1, -3) → Q''(1, 3)
- R'(-2, -6) → R''(-2, 6)
- S'(-1, -1) → S''(-1, 1)

✔ So:
- Q''(1, 3)
- R''(-2, 6)
- S''(-1, 1)

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Problem 3: Rotate △CAR 180° about the origin

Given:
- C(-1, -4)
- A(2, 3)
- R(-3, -2)

Rotation 180° about the origin: (x, y) → (-x, -y)

- C': (-(-1), -(-4)) = (1, 4)
- A': (-2, -3)
- R': (3, 2)

✔ So:
- C'(1, 4)
- A'(-2, -3)
- R'(3, 2)

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Problem 4: Reflect △C'A'R' over the line y = x

Now reflect the rotated triangle over the line y = x.

Reflection over y = x: (x, y) → (y, x)

From Problem 3:
- C'(1, 4) → C''(4, 1)
- A'(-2, -3) → A''(-3, -2)
- R'(3, 2) → R''(2, 3)

✔ So:
- C''(4, 1)
- A''(-3, -2)
- R''(2, 3)

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Problem 5: Analysis & Multiple Transformations

#### Part A: What did you notice in problems 1&2 and 3&4? How were the shapes related?

Let’s compare:

- In Problems 1 & 2, we:
- Translated △QRS → △Q'R'S'
- Then reflected △Q'R'S' over x-axis → △Q''R''S''

- In Problems 3 & 4, we:
- Rotated △CAR 180° → △C'A'R'
- Then reflected over y = x → △C''A''R''

But look at the final coordinates:

From Problem 4:
- C''(4,1), A''(-3,-2), R''(2,3)

Wait — these are the same as original points of △QRS!

Original △QRS:
- Q(4,1), R(1,-2), S(2,3)

But our final points are:
- C''(4,1), A''(-3,-2), R''(2,3)

So:
- C'' = Q
- R'' = S
- But A'' = (-3,-2), which is not a point in △QRS

Wait — let's check again.

Actually, in Problem 3, we had:
- C(-1,-4), A(2,3), R(-3,-2)

After 180° rotation:
- C'(1,4), A'(-2,-3), R'(3,2)

Then reflection over y=x:
- C''(4,1), A''(-3,-2), R''(2,3)

Now compare to original △QRS:
- Q(4,1), R(1,-2), S(2,3)

So:
- C'' = Q(4,1)
- R'' = S(2,3)
- A'' = (-3,-2) → this is not R or S

But wait — original R is (1,-2), but here A'' is (-3,-2)

Hmm — perhaps no direct match.

But let's go back to Problem 1 and 2:

We started with △QRS: Q(4,1), R(1,-2), S(2,3)

After translation: Q'(1,-3), R'(-2,-6), S'(-1,-1)

Then reflection over x-axis: Q''(1,3), R''(-2,6), S''(-1,1)

So final image has points: (1,3), (-2,6), (-1,1)

Now, Problem 3 and 4 ended with: (4,1), (-3,-2), (2,3)

So different.

But wait — maybe there's a pattern?

Wait! Let’s look at Problem 4’s final points:
- C''(4,1), A''(-3,-2), R''(2,3)

Compare to original △QRS: Q(4,1), R(1,-2), S(2,3)

So:
- C'' = Q
- R'' = S
- A'' = (-3,-2) → not matching

But if we look at Problem 1: we translated QRS → Q’R’S’ → then reflected → Q''R''S''

And in Problem 4, we got C''(4,1), A''(-3,-2), R''(2,3)

But notice: C''(4,1) is same as Q(4,1)

R''(2,3) is same as S(2,3)

A''(-3,-2) — is this related?

Wait — what if we consider that rotating 180° then reflecting over y=x is equivalent to some other transformation?

But perhaps the key idea is order of operations.

Let’s re-read the last part of Question 5:

> "How would you rotate △CAR 180° about the origin and then reflect it over the line y = x?"

We already did that in Problems 3 & 4.

But let’s answer directly.

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✔ Final Answer to Question 5:

#### Part A: Observations

In problems 1 & 2:
- We translated △QRS by (x–3, y–4), then reflected over the x-axis.
- The shape was moved and flipped vertically.

In problems 3 & 4:
- We rotated △CAR 180° about the origin, then reflected over y = x.
- The shape was rotated (flipped around the origin) and then mirrored across the diagonal line y = x.

The shapes are related because both involve combinations of rigid transformations (translation, rotation, reflection), preserving size and shape. However, the positions and orientations changed differently.

But here’s an important observation:

Let’s see what happens when we translate first, then reflect.

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#### Part B: Translate △QRS by (x–3, y–4), then reflect over x-axis

We already did this:

Step 1: Translate:
- Q(4,1) → Q'(1,-3)
- R(1,-2) → R'(-2,-6)
- S(2,3) → S'(-1,-1)

Step 2: Reflect over x-axis:
- Q''(1,3)
- R''(-2,6)
- S''(-1,1)

So final image has vertices: (1,3), (-2,6), (-1,1)

This is the final image after both transformations.

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#### Part C: How to rotate △CAR 180° about the origin and then reflect over y = x?

We already did this in Problems 3 and 4.

Steps:

1. Rotate 180° about origin: (x,y) → (-x,-y)
- C(-1,-4) → C'(1,4)
- A(2,3) → A'(-2,-3)
- R(-3,-2) → R'(3,2)

2. Reflect over y = x: (x,y) → (y,x)
- C'(1,4) → C''(4,1)
- A'(-2,-3) → A''(-3,-2)
- R'(3,2) → R''(2,3)

Final image: C''(4,1), A''(-3,-2), R''(2,3)

Note: This final triangle has points that match the original Q(4,1) and S(2,3), but A'' is new.

But interestingly, if we reverse the order, we might get different results.

But for this problem, we’ve answered:

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✔ Summary of Answers:

#### 1) Translation:
- Q'(1, -3)
- R'(-2, -6)
- S'(-1, -1)

#### 2) Reflection over x-axis:
- Q''(1, 3)
- R''(-2, 6)
- S''(-1, 1)

#### 3) Rotation 180° about origin:
- C'(1, 4)
- A'(-2, -3)
- R'(3, 2)

#### 4) Reflection over y = x:
- C''(4, 1)
- A''(-3, -2)
- R''(2, 3)

#### 5) Final Answers:

What did you notice?

- In problems 1&2: We moved the triangle left/down, then flipped it upside down (over x-axis).
- In problems 3&4: We rotated the triangle 180° (which flips it and centers it at origin), then mirrored it over the diagonal y = x.

These transformations show how multiple rigid motions can change position and orientation while preserving size and shape.

Translate △QRS by (x–3, y–4), then reflect over x-axis:

- Final image: Q''(1,3), R''(-2,6), S''(-1,1)

Rotate △CAR 180° about origin, then reflect over y = x:

- Step 1: Rotate → C'(1,4), A'(-2,-3), R'(3,2)
- Step 2: Reflect over y=x → C''(4,1), A''(-3,-2), R''(2,3)

So the final image ends up at:
- C''(4,1), A''(-3,-2), R''(2,3)

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🟩 Final Boxed Answers:

1)
Q'(1, -3)
R'(-2, -6)
S'(-1, -1)

2)
Q''(1, 3)
R''(-2, 6)
S''(-1, 1)

3)
C'(1, 4)
A'(-2, -3)
R'(3, 2)

4)
C''(4, 1)
A''(-3, -2)
R''(2, 3)

5)
- After translating △QRS by (x–3, y–4) and reflecting over x-axis, the final image has vertices: (1,3), (-2,6), (-1,1).
- To rotate △CAR 180° about origin and reflect over y=x:
First rotate: (x,y) → (-x,-y), then reflect: (x,y) → (y,x).
Final image: C''(4,1), A''(-3,-2), R''(2,3).

✔ All steps completed!