Transformed Figures Worksheet Collection For Teaching & Learning - Free Printable

Let's solve each of the four problems on the worksheet step by step. We are to determine if the two figures in each coordinate plane are congruent, and if so, describe the congruence transformations (such as translations, reflections, rotations) that map Figure 1 onto Figure 2.

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🔷 Problem 1 (Top Left)

Figures:
- Figure 1: Green L-shaped figure with vertices at (1,4), (1,2), (3,2), (3,1), (2,1)
- Figure 2: Orange L-shaped figure with vertices at (-3,2), (-3,0), (-1,0), (-1,-1), (-2,-1)

#### ✔ Step 1: Are they congruent?
Both are L-shaped figures made of 5 unit squares. They have the same shape and size — just positioned differently.

✔ Yes, they are congruent.

#### ✔ Step 2: Transformation from Figure 1 to Figure 2
We can compare corresponding points:
- (1,4) → (-3,2): Δx = -4, Δy = -2
- (1,2) → (-3,0): Δx = -4, Δy = -2
- (3,2) → (-1,0): Δx = -4, Δy = -2
- (3,1) → (-1,-1): Δx = -4, Δy = -2
- (2,1) → (-2,-1): Δx = -4, Δy = -2

All points shift by (-4, -2).

👉 So, the transformation is a translation of 4 units left and 2 units down.

> ✔ Answer: The figures are congruent.
> Transformation: Translate Figure 1 by (x – 4, y – 2) to map it onto Figure 2.

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🔷 Problem 2 (Top Right)

Figures:
- Figure 1: Green triangle with vertices at (-3,-4), (-1,-4), (-2,-2)
- Figure 2: Orange triangle with vertices at (1,5), (2,3), (1,3)

#### ✔ Step 1: Are they congruent?

Let’s check side lengths:

Figure 1:
- A(-3,-4), B(-1,-4), C(-2,-2)
- AB = 2 units (horizontal)
- AC = √[(-2+3)² + (-2+4)²] = √[1 + 4] = √5
- BC = √[(-1+2)² + (-4+2)²] = √[1 + 4] = √5

So it's an isosceles triangle with sides: 2, √5, √5

Figure 2:
- D(1,5), E(2,3), F(1,3)
- DE = √[(2-1)² + (3-5)²] = √[1 + 4] = √5
- DF = 2 units (vertical)
- EF = √[(2-1)² + (3-3)²] = 1? Wait: (2,3) to (1,3) = 1 unit horizontally.

Wait — this seems inconsistent.

Wait! Let's list the correct vertices:

Looking closely:
- Figure 2: (1,5), (2,3), (1,3)
- So: D(1,5), E(2,3), F(1,3)

Now compute:
- DE: √[(2-1)² + (3-5)²] = √[1 + 4] = √5
- DF: |5 - 3| = 2 (vertical)
- EF: |2 - 1| = 1 (horizontal)

But wait — this gives sides: 2, 1, √5 — which is not the same as Figure 1 (which had 2, √5, √5).

But let's double-check the shapes.

Actually, both triangles appear to be right triangles with legs of length 2 and 2?

Wait — let’s re-express:

Figure 1:
- From (-3,-4) to (-1,-4): horizontal leg = 2
- From (-1,-4) to (-2,-2): not vertical or horizontal

Wait — better to plot or calculate distances.

Let me do exact calculations:

Figure 1:
- A(-3,-4), B(-1,-4), C(-2,-2)
- AB: distance = √[(-1+3)² + (-4+4)²] = √[4] = 2
- AC: √[(-2+3)² + (-2+4)²] = √[1 + 4] = √5
- BC: √[(-1+2)² + (-4+2)²] = √[1 + 4] = √5

So sides: 2, √5, √5 → isosceles

Figure 2:
- D(1,5), E(2,3), F(1,3)
- DF: from (1,5) to (1,3): vertical = 2 units
- EF: from (2,3) to (1,3): horizontal = 1 unit
- DE: √[(2-1)² + (3-5)²] = √[1 + 4] = √5

Sides: 2, 1, √5 → different from Figure 1

But wait — this suggests not congruent?

But visually, they look similar.

Wait — maybe I misread the coordinates.

Look again:

Figure 2 has points:
- (1,5), (2,3), (1,3)

Wait — (1,5) to (1,3) = 2 units down
- (1,3) to (2,3) = 1 unit right
- (2,3) to (1,5): diagonal

But Figure 1:
- (-3,-4) to (-1,-4): 2 units right
- (-1,-4) to (-2,-2): up-left
- (-2,-2) to (-3,-4): down-left

Wait — actually, let’s see if there's a rotation or reflection.

Try comparing angles.

Alternatively, try to see if one is a rotation/reflection of the other.

Let’s suppose we rotate Figure 1 90° clockwise around origin.

But better: let's consider possible transformation.

Wait — notice that both seem to be right triangles?

Let’s check angles.

In Figure 1:
- At point B(-1,-4): vectors BA = (-2,0), BC = (-1,2)
- Dot product: (-2)(-1) + (0)(2) = 2 ≠ 0 → not right angle

Wait — maybe not.

Alternatively, perhaps I should count grid squares.

But here's a better idea: compare areas and side lengths.

Figure 1:
- It's a triangle with base 2 (from x=-3 to -1 at y=-4), height 2 (up to y=-2), but not a right triangle?

Wait — actually, plotting:
- Points: (-3,-4), (-1,-4), (-2,-2)

This forms a triangle where:
- From (-3,-4) to (-1,-4): 2 units right
- From (-1,-4) to (-2,-2): 1 left, 2 up
- From (-2,-2) to (-3,-4): 1 left, 2 down

Wait — no, it's not symmetric.

But let’s calculate area using shoelace formula.

Shoelace for Figure 1:
Points: (-3,-4), (-1,-4), (-2,-2), back to (-3,-4)

Sum1 = (-3)(-4) + (-1)(-2) + (-2)(-4) = 12 + 2 + 8 = 22
Sum2 = (-4)(-1) + (-4)(-2) + (-2)(-3) = 4 + 8 + 6 = 18
Area = |22 - 18| / 2 = 4/2 = 2 square units

Figure 2:
Points: (1,5), (2,3), (1,3)

Shoelace:
(1,5), (2,3), (1,3), (1,5)

Sum1 = 1×3 + 2×3 + 1×5 = 3 + 6 + 5 = 14
Sum2 = 5×2 + 3×1 + 3×1 = 10 + 3 + 3 = 16
Area = |14 - 16| / 2 = 2/2 = 1 square unit

✘ Area of Figure 1 = 2, Figure 2 = 1 → Not congruent

But wait — that can't be. Visually, they look the same size.

Wait — did I make a mistake?

Let’s recompute Figure 2:

Points: (1,5), (2,3), (1,3)

List in order:
- (1,5)
- (2,3)
- (1,3)
- back to (1,5)

Shoelace:

Sum1 = (1)(3) + (2)(3) + (1)(5) = 3 + 6 + 5 = 14
Sum2 = (5)(2) + (3)(1) + (3)(1) = 10 + 3 + 3 = 16
Area = |14 - 16| / 2 = 1 → 1 square unit

Figure 1: earlier I got 2.

But let’s recalculate Figure 1:

Points: (-3,-4), (-1,-4), (-2,-2)

Sum1 = (-3)(-4) + (-1)(-2) + (-2)(-4) = 12 + 2 + 8 = 22
Sum2 = (-4)(-1) + (-4)(-2) + (-2)(-3) = 4 + 8 + 6 = 18
Area = |22 - 18| / 2 = 4/2 = 2

But Figure 2 has area 1 → not congruent

But wait — visually, both triangles appear to have the same size?

Wait — let's count the grid.

Figure 2:
- From (1,5) to (2,3): down 2, right 1
- From (2,3) to (1,3): left 1
- From (1,3) to (1,5): up 2

So it's a triangle with base 1 and height 2 → area = (1 × 2)/2 = 1

Figure 1:
- Base from (-3,-4) to (-1,-4): length 2
- Height from y=-4 to y=-2: 2 units, but apex is at (-2,-2), which is above the base

So it's a triangle with base 2, height 2 → area = (2 × 2)/2 = 2

So yes — areas differ: 2 vs 1 → not congruent

But wait — that can’t be right because the green triangle looks larger than orange?

Wait — actually, looking at the image:

- Figure 1 (green): bottom at y=-4, top at y=-2 → height 2
- Width: from x=-3 to x=-1 → width 2
- But it's not a full triangle; it's a triangle with vertices at (-3,-4), (-1,-4), (-2,-2)

This is a triangle with base 2 and height 2, but the apex is centered → area = (base × height)/2 = (2 × 2)/2 = 2

Figure 2 (orange): points (1,5), (2,3), (1,3)
- This is a triangle with base from (1,3) to (2,3): length 1
- Height from y=3 to y=5: 2 units → area = (1 × 2)/2 = 1

So areas are different → not congruent

But wait — the orange triangle looks like it might be the same size?

Wait — let’s measure the sides.

Figure 1:
- AB = 2
- AC = √[(−2+3)² + (−2+4)²] = √[1 + 4] = √5
- BC = √[(−1+2)² + (−4+2)²] = √[1 + 4] = √5

So sides: 2, √5, √5

Figure 2:
- DF = 2 (vertical)
- EF = 1 (horizontal)
- DE = √[(2−1)² + (3−5)²] = √[1 + 4] = √5

So sides: 2, 1, √5 → different from Figure 1

So not congruent

But wait — could it be rotated?

No, because side lengths don't match.

Therefore:

> ✘ Not congruent — different side lengths and areas.

But wait — maybe I misidentified the vertices?

Let’s double-check the orange triangle.

From the image: Figure 2 is a triangle pointing upward-right, with:
- Top vertex at (1,5)
- Bottom-left at (1,3)
- Bottom-right at (2,3)

So yes — it's a right triangle with legs 2 (vertical) and 1 (horizontal) → area = 1

Green triangle: base 2, height 2, but not a right triangle — it's isosceles with two equal sides of √5 and base 2 → area = 2

So not congruent

> ✔ Answer: The figures are not congruent because they have different sizes (different areas and side lengths). Figure 1 has area 2, Figure 2 has area 1.

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🔷 Problem 3 (Bottom Left)

Figures:
- Figure 1: Green triangle with vertices at (1,5), (3,5), (2,2)
- Figure 2: Orange triangle with vertices at (-2,-3), (-1,-5), (-3,-5)

#### ✔ Step 1: Are they congruent?

Let’s compute side lengths.

Figure 1:
- A(1,5), B(3,5), C(2,2)
- AB = 2 units (horizontal)
- AC = √[(2-1)² + (2-5)²] = √[1 + 9] = √10
- BC = √[(3-2)² + (5-2)²] = √[1 + 9] = √10

So: sides 2, √10, √10 → isosceles

Figure 2:
- D(-2,-3), E(-1,-5), F(-3,-5)
- DE = √[(-1+2)² + (-5+3)²] = √[1 + 4] = √5
- DF = √[(-3+2)² + (-5+3)²] = √[1 + 4] = √5
- EF = |-1 - (-3)| = 2 units

So sides: 2, √5, √5 → different from Figure 1

Wait — not matching.

But wait — maybe I have the wrong points.

Let’s look carefully:

Figure 2: orange triangle at bottom left
- Left vertex: (-3,-5)
- Right vertex: (-1,-5)
- Top vertex: (-2,-3)

So:
- D(-3,-5), E(-1,-5), F(-2,-3)

Now:
- DE = 2 units (horizontal)
- DF = √[(-2+3)² + (-3+5)²] = √[1 + 4] = √5
- EF = √[(-1+2)² + (-5+3)²] = √[1 + 4] = √5

So sides: 2, √5, √5

But Figure 1 has sides: 2, √10, √10

→ Not the same → not congruent

But wait — areas?

Figure 1: base 2, height 3 → area = (2 × 3)/2 = 3

Figure 2: base 2, height 2 → area = (2 × 2)/2 = 2 → different

So not congruent

Wait — but visually, they look similar?

Wait — Figure 1 has height from y=2 to y=5 → 3 units
Figure 2 has height from y=-5 to y=-3 → 2 units

So different sizes → not congruent

> ✔ Answer: The figures are not congruent because they have different sizes (different side lengths and areas).

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🔷 Problem 4 (Bottom Right)

Figures:
- Figure 1: Green L-shape at bottom right: vertices at (1,-5), (1,-3), (2,-3), (2,-1), (3,-1), (3,-2), (1,-2)? Wait — better to trace.

From image:
- Figure 1: green shape with blocks at:
- (1,-5), (2,-5), (3,-5)
- (1,-4), (2,-4)
- (1,-3), (2,-3)
- (1,-2), (2,-2)
- (1,-1), (2,-1)
- (3,-1)

Wait — actually, looking at the image:
- Vertical column from (1,-5) to (1,-1): 5 units
- Then at (2,-5), (2,-4), (2,-3), (2,-2), (2,-1): 5 units
- Then at (3,-1): one block

So total: 5 + 5 + 1 = 11 squares?

Wait — let’s count:

- Column at x=1: y=-5 to -1 → 5 blocks
- Column at x=2: y=-5 to -1 → 5 blocks
- Block at (3,-1) → 1 block

Total: 11 blocks

But Figure 2: orange shape at top left
- x=-5 to -2: y=4 to 5
- Blocks:
- (-5,4), (-4,4), (-3,4), (-2,4)
- (-5,5), (-4,5), (-3,5), (-2,5)
- (-3,5) already counted
- And (-3,4) etc.

Wait — let’s count:

- Row y=5: x=-5 to -2 → 4 blocks
- Row y=4: x=-5 to -2 → 4 blocks
- Row y=3: only x=-3 → 1 block

Total: 4 + 4 + 1 = 9 blocks

But Figure 1 has more blocks.

Wait — no, let’s recount Figure 1:

From image:
- Figure 1: green shape
- Vertical line at x=1: y=-5, -4, -3, -2, -1 → 5 blocks
- Vertical line at x=2: y=-5, -4, -3, -2, -1 → 5 blocks
- One block at (3,-1)

Total: 5 + 5 + 1 = 11 blocks

Figure 2: orange shape
- Row y=5: x=-5, -4, -3, -2 → 4
- Row y=4: x=-5, -4, -3, -2 → 4
- Row y=3: x=-3 → 1
- Total: 9 blocks

So different number of blocks → not congruent

But wait — maybe I miscounted.

Wait — Figure 2 has:
- ( -5,5), (-4,5), (-3,5), (-2,5) → 4
- ( -5,4), (-4,4), (-3,4), (-2,4) → 4
- ( -3,3) → 1
- Total: 9

Figure 1:
- (1,-5), (1,-4), (1,-3), (1,-2), (1,-1) → 5
- (2,-5), (2,-4), (2,-3), (2,-2), (2,-1) → 5
- (3,-1) → 1
- Total: 11

So not congruent due to different sizes.

But wait — perhaps they are supposed to be the same?

Wait — maybe Figure 1 is missing something.

Wait — no, based on the image, Figure 1 has 11 blocks, Figure 2 has 9 → not congruent

But let’s double-check the image.

Wait — actually, in Figure 1, is (3,-1) the only extra? Yes.

But in Figure 2, is there a block at (-3,3)? Yes.

So 9 vs 11 → not congruent

> ✔ Answer: The figures are not congruent because they have different numbers of unit squares (11 vs 9), so they are not the same size.

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✔ Final Answers Summary:

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Problem 1 (Top Left):
- Congruent? Yes
- Transformation: Translate Figure 1 by (x – 4, y – 2)

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Problem 2 (Top Right):
- Congruent? No
- Reason: Different side lengths and areas (2 vs 1). Figure 1 has sides 2, √5, √5; Figure 2 has sides 2, 1, √5.

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Problem 3 (Bottom Left):
- Congruent? No
- Reason: Different side lengths and areas. Figure 1 has sides 2, √10, √10; Figure 2 has sides 2, √5, √5.

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Problem 4 (Bottom Right):
- Congruent? No
- Reason: Different numbers of unit squares (11 vs 9), so not the same size.

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