a transformation is shown in the diagram. ∠a ≅ ∠a and ∠c ≅ ∠c
which transformation is shown in the figure?
○ δabc was dilated to form δabc. the transformation is not isometric.
○ δabc was reflected to form δabc. the transformation is isometric.
○ δabc was rotated to form δabc. the transformation is isometric.
○ δabc was translated to form δabc. the transformation is not isometric.

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# Brief Explanations:
1. Recall the properties of transformations:
   - **Dilation**: Changes the size (scale factor ≠ 1), not isometric (preserves distance/angle? No, size changes). But angles remain congruent.
   - **Reflection**: Flips over a line, isometric (preserves distance, angle, shape/size).
   - **Rotation**: Turns around a point, isometric (preserves distance, angle, shape/size).
   - **Translation**: Slides, isometric (preserves distance, angle, shape/size).
2. Analyze the diagram:
   - The triangles have congruent angles (∠A ≅ ∠A', ∠C ≅ ∠C'), and the markings (tick marks) suggest corresponding sides are congruent (so shape/size preserved? Wait, no—wait, the orientation and position: Wait, the first triangle (ABC) has a right angle at B, the second (A'B'C') has a right angle at B'. The sides: Let's check the tick marks. In ABC: AB has two ticks, AC has one tick, BC has one tick? Wait, no, maybe I misread. Wait, the first triangle: AB (top) has two ticks, AC (left) has one tick, BC (right) has one tick? Wait, the second triangle: A'C' has one tick, B'C' has one tick, A'B' has two ticks? Wait, no, maybe the transformation is a rotation? Wait, no—wait, the key is: Isometric transformations (reflection, rotation, translation) preserve size and shape (so congruent triangles). Dilation changes size (similar triangles, not congruent).
   - Wait, the first option says "dilated" (so not isometric) but angles are congruent (which is true for dilation, as dilation preserves angles). But wait, the tick marks: If the sides have the same number of ticks, that would mean congruent sides. Wait, maybe I made a mistake. Wait, let's re-express:
   - **Reflection**: Flips, so the image is a mirror image. But here, the triangle is rotated? Wait, no—wait, the right angle is at B in ABC and at B' in A'B'C', but the positions: ABC has A---B (horizontal), B---C (vertical down), A---C (slanted). A'B'C' has C'---B' (horizontal), B'---A' (vertical up), C'---A' (slanted). So this looks like a rotation (maybe 90 degrees or some angle) or a reflection? Wait, no—wait, the correct answer: Let's check the options:
   - Option 1: Dilated (not isometric). But if the sides have the same tick marks, that would mean congruent, so dilation (which changes size) would not have congruent sides. So maybe the tick marks are for corresponding sides, but maybe the diagram shows similar triangles? Wait, no, the angles are congruent, and if it's a dilation, the sides would be proportional, not congruent. But the tick marks suggest congruent sides. Wait, maybe I misinterpret the tick marks. Alternatively, maybe the correct answer is the first option? No, wait:
   - Wait, isometric transformations (reflection, rotation, translation) preserve distance (so sides are congruent), so the image is congruent to the pre-image. Dilation preserves angles but not distances (so image is similar, not congruent).
   - Now, looking at the diagram: The first triangle (ABC) and the second (A'B'C')—do they look congruent? The tick marks: In ABC, AB has two ticks, AC has one tick, BC has one tick? Wait, no, maybe AB (top) has two ticks, BC (right) has one tick, AC (left) has one tick. In A'B'C', A'C' has one tick, B'C' has one tick, A'B' has two ticks. So corresponding sides: AB ↔ A'B' (two ticks), AC ↔ A'C' (one tick), BC ↔ B'C' (one tick). So sides are congruent. So it's an isometric transformation? But the first option says "dilated" (not isometric), which would mean sides are not congruent. So that's a contradiction. Wait, maybe the diagram is misread. Alternatively, maybe the correct answer is the first option? No, wait:
   - Wait, the problem says "∠A ≅ ∠A' and ∠C ≅ ∠C'". So angles are congruent. Now, let's check each option:
   - Option 1: Dilated (so scale factor ≠ 1, not isometric). Angles are congruent (true for dilation). But if sides are congruent (tick marks), then scale factor is 1, which is not dilation. So maybe the tick marks are for proportional sides? No, tick marks are for congruent. So maybe the first option is wrong.
   - Option 2: Reflected. Reflection is isometric. But does the diagram look like a reflection? A reflection would flip over a line, so the orientation would be mirrored. But here, the triangle is rotated? Wait, maybe the diagram is a rotation. Wait, option 3: Rotated (isometric). Rotation preserves angles and sides, so isometric.
   - Wait, maybe I made a mistake. Let's re-express:
   - Isometric transformations: reflection, rotation, translation. These preserve length (so sides congruent) and angle (so angles congruent). Dilation preserves angle but not length (so sides proportional, angles congruent).
   - Now, the options:
   - Option 1: Dilated (not isometric) → but if sides are congruent, dilation would have scale factor 1, which is not dilation. So this is wrong.
   - Option 2: Reflected (isometric) → but does the diagram show a reflection? Let's see: ABC has right angle at B, A'B'C' has right angle at B'. The position: ABC is A---B (horizontal), B---C (vertical down), A---C (slanted left-down). A'B'C' is C'---B' (horizontal), B'---A' (vertical up), C'---A' (slanted left-up). So this looks like a rotation (90 degrees clockwise or counterclockwise) or a reflection? Wait, a reflection over a vertical or horizontal line? Maybe not. Alternatively, a rotation.
   - Option 3: Rotated (isometric) → rotation preserves congruence, so isometric. This makes sense.
   - Option 4: Translated (not isometric) → translation is isometric, so this is wrong (says "not isometric").
   - Wait, but option 3 says "rotated to form ΔA'B'C'. The transformation is isometric." That's correct, because rotation is isometric (preserves distance and angle).
   - Wait, but let's check again. The first triangle: right angle at B, legs AB (horizontal) and BC (vertical), hypotenuse AC. The second triangle: right angle at B', legs B'C' (horizontal) and B'A' (vertical), hypotenuse A'C'. So it's a rotation (maybe 90 degrees) of the first triangle to get the second. So rotation is isometric, so option 3 is correct. Wait, but earlier I thought maybe option 1, but no—dilation would change size, but the tick marks suggest congruent sides. So the correct answer is option 3? Wait, no, wait the first option: "ΔABC was dilated to form ΔA'B'C'. The transformation is not isometric." But if the sides are congruent, dilation would have scale factor 1, which is not dilation. So maybe the tick marks are for proportional sides? No, tick marks are for congruent. So perhaps the diagram is misrepresented, but based on the options:
   - Wait, maybe the correct answer is the first option? No, because dilation is not isometric (changes size), but if angles are congruent and sides are proportional, but the tick marks suggest congruent. Alternatively, maybe the answer is option 3: rotated, isometric.
   - Wait, let's recall: Isometric transformations (rigid motions) are reflection, rotation, translation. They preserve distance (so sides congruent) and angle (so angles congruent). Dilation is not isometric (preserves angle, not distance). So if the triangles are congruent (sides congruent, angles congruent), then it's a rigid motion (isometric). So options 2,3,4 are isometric (but option 4 says "not isometric", which is wrong). So between 2,3: reflection or rotation. The diagram shows the triangle rotated (not flipped), so rotation. So option 3: "ΔABC was rotated to form ΔA'B'C'. The transformation is isometric." is correct.

# Answer:
$\boldsymbol{	ext{ΔABC was rotated to form ΔA'B'C'. The transformation is isometric.}}$ (The third option)

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