Congruent Figures Practice Worksheet A from Mathcation.com.

Let's solve the problems on this "Congruent Figures Practice Worksheet A" step by step.

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Section 1: Determine if the figures below are Congruent

> Congruent figures have the same shape and size. They can be rotated, reflected, or translated, but their side lengths and angles must match exactly.

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#### Problem 1)
Two right triangles:

- The first triangle is a right triangle with legs aligned vertically and horizontally.
- The second triangle is also a right triangle, but it's rotated (appears to be flipped or rotated 90°).

✔ Are they congruent?
Yes — both appear to have the same side lengths and angles. One is just rotated.
👉 Answer: YES, they are congruent.

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#### Problem 2)
A large rectangle and a small rectangle.

- The first rectangle is clearly larger than the second.
- They are similar in shape but not the same size.

✘ Are they congruent?
No — different sizes.
👉 Answer: NO, they are not congruent.

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#### Problem 3)
Two arrow shapes:

- Both arrows are identical in shape and size.
- One appears slightly smaller? Wait — let’s examine closely.

Wait — actually, looking at the image:
- The left arrow is taller and wider than the right arrow.
- So although they look similar, the sizes differ.

✘ Are they congruent?
No — one is scaled down.
👉 Answer: NO, they are not congruent.

> ⚠️ Note: If they were the same size and shape (even if rotated), they’d be congruent. But here, the right arrow is smaller.

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✔ Summary for Section 1:

1) Yes – congruent
2) No – not congruent (different sizes)
3) No – not congruent (different sizes)

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Section 2: Describe the sequence of transformations from Figure A to Figure A'

We need to describe how Figure A is transformed into Figure A' using translations, rotations, reflections, or dilations.

Let’s go through each one.

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#### Problem 1)

- Figure A is on the left side of the y-axis.
- Figure A' is on the right side, mirrored across the y-axis.

✔ Observation:
The figure is reflected over the y-axis.

👉 Answer: Reflect Figure A over the y-axis.

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#### Problem 2)

- Figure A is on the left.
- Figure A' is on the right, but rotated.

Let’s check:
- It looks like Figure A is rotated 90° clockwise and then possibly translated.

But wait — actually, compare coordinates roughly:
- Original A has a "point" facing up-left.
- A' has a point facing down-right → suggests a rotation.

Looking more carefully:
- The orientation changed significantly.
- The shape is flipped and turned.

Actually, let’s think: From A to A', it seems like a reflection over the x-axis, then a translation to the right, or maybe a rotation?

Wait — better way: Let's assume grid units.

Suppose we pick a vertex of A and see where it goes in A'.

But since it's hard without exact coordinates, visually:

- A is on the left, pointing upward.
- A' is on the right, pointing downward.

This looks like a rotation of 180° about the origin.

Let’s test: If you rotate A 180° around the origin, it would end up on the opposite side, inverted.

Yes — that matches.

👉 Answer: Rotate Figure A 180° about the origin.

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#### Problem 3)

- Figure A is in the bottom-right quadrant.
- Figure A' is in the top-right quadrant.

It looks like:
- The triangle is rotated and moved.

Check:
- The original triangle is pointing up-right.
- The image triangle is pointing down-left? No — actually, it’s pointing up-left.

Wait — let’s analyze:

- Figure A: base on bottom, point upward.
- Figure A': base on top, point downward → so it’s inverted.

Also, it’s shifted up and to the left.

So likely: Reflection over x-axis, then translate up and left?

But let’s consider: Is it a rotation?

Alternatively, observe:
If you reflect over the x-axis, the triangle flips upside down. Then move it.

But wait — the position: A is near (4, -2), A' is near (2, 2). So it moved up and left.

But reflection over x-axis would flip it vertically.

Let’s suppose:
- Reflect over x-axis → flips it upside down.
- Then translate left 2 units and up 4 units.

That might work.

But another possibility: Rotation 180° about a point?

Try rotating 180° about the origin:
- Point (x,y) → (-x,-y)
- So (4,-2) → (-4,2) → but A' is at (2,2), so no.

Not matching.

Wait — maybe a reflection over the line y = x? Unlikely.

Alternative idea: Reflection over the x-axis, then translate.

But simpler: Look at orientation.

Original triangle has vertical leg on the left, horizontal base.

Image triangle has vertical leg on the right, horizontal base — so it's reflected over the y-axis, but also flipped?

Wait — perhaps it's a rotation of 90° counterclockwise?

Try that:
- A point at (4, -2): rotate 90° CCW → becomes (2, 4) → which is close to A’ (around (2,2)) — not quite.

Wait — maybe reflect over the x-axis, then translate up?

But let’s do this carefully.

Better approach: Count grid squares.

Assume:
- Bottom-left vertex of A is at (3, -2)
- Top-left at (3, -1)
- Bottom-right at (4, -2)

Then A' has:
- Bottom-left at (2, 2)
- Top-left at (2, 3)
- Bottom-right at (3, 2)

Now compare:
- A: (3,-2) → A': (2,2): Δx = -1, Δy = +4
- A: (3,-1) → A': (2,3): Δx = -1, Δy = +4
- A: (4,-2) → A': (3,2): Δx = -1, Δy = +4

So every point moves left 1 unit, up 4 units — that’s a translation.

But the shape is flipped — the triangle is now pointing up, but in A it was pointing up too?

Wait — in A, the point is at (3,-1), so it points up.

In A', the point is at (2,3), so it points up — same direction.

But the sides: In A, the right side is vertical; in A', the left side is vertical.

So it's mirrored.

So it’s not just translation — it’s a reflection followed by translation?

Wait — look again: the triangle is oriented differently.

From A to A':
- The hypotenuse goes from bottom-left to top-right in A.
- In A', it goes from bottom-left to top-right — same slope.

Wait — actually, it’s not reflected — it’s just rotated and translated?

Let’s try rotation.

Rotate A 90° clockwise about origin:
- (x,y) → (y, -x)
- (3,-2) → (-2, -3) → not matching.

Rotate 90° counterclockwise: (x,y) → (-y,x)
- (3,-2) → (2,3) → yes!
- (3,-1) → (1,3) → but A' has (2,3), (2,2), (3,2) — doesn't match.

Wait — maybe not.

Alternative idea: Reflect over the line y = -x?

Too complex.

Let’s re-express:

Actually, upon closer inspection:
- Figure A: a right triangle with legs along the bottom and left side.
- Figure A': a right triangle with legs along the top and right side.

So it’s been rotated 180°?

No — 180° would make it upside down.

But here, it’s upright, but flipped.

Wait — actually, it's reflected over the x-axis, then translated?

No — if reflected over x-axis, the point (3,-1) becomes (3,1), but A' has (2,3).

Wait — perhaps it’s a reflection over the line y = x?

(3,-2) → (-2,3) — not matching.

I think there's a mistake in my analysis.

Let me try again.

Look at the two triangles:
- A: vertices approximately at (3,-2), (3,-1), (4,-2)
- A': vertices at (2,2), (2,3), (3,2)

So:
- A: right angle at (3,-2)
- A': right angle at (2,2)

So:
- From A to A':
- (3,-2) → (2,2): Δx = -1, Δy = +4
- (3,-1) → (2,3): Δx = -1, Δy = +4
- (4,-2) → (3,2): Δx = -1, Δy = +4

All points move left 1, up 4 → so translation.

But the shape is not rotated or reflected — it’s just moved.

But wait — the orientation is different?

No — in A, the right angle is at bottom-left, and the triangle points up-right.

In A', the right angle is at bottom-left, and it points up-right — same orientation.

Wait — but the labels: A' is labeled inside, but the triangle is the same.

Wait — actually, the triangle in A' is identical in shape and orientation, just shifted.

So it’s a translation.

But why does it look like it’s rotated?

Wait — no — the triangle is not rotated — it’s just moved.

But look at the graph: the triangle in A is pointing up, and A' is pointing up — same direction.

And the movement is consistent: all points shift left 1, up 4.

So the transformation is:

👉 Translate Figure A 1 unit left and 4 units up.

✔ That’s correct.

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#### Problem 4)

- Figure A is a rectangle at the bottom, centered around (0, -5).
- Figure A' is a rectangle at the top, centered around (0, 5).

So:
- It’s moved from y = -5 to y = 5 → total change of +10 units in y-direction.
- No change in x-direction.

Also, the shape is the same, no rotation or reflection.

So:

👉 Translate Figure A 10 units up.

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✔ Final Answers:

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Section 1: Are the figures congruent?

1) Yes
2) No
3) No

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Section 2: Describe the transformation from A to A'

1) Reflect over the y-axis
2) Rotate 180° about the origin
3) Translate 1 unit left and 4 units up
4) Translate 10 units up

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📌 Summary of Key Concepts:

- Congruent: Same shape and size (can be moved via rigid transformations: translation, rotation, reflection).
- Rigid transformations preserve size and shape.
- Dilation changes size → not congruent unless scale factor is 1.

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