Figure Analogy - Advanced Level: visual reasoning ADVANCED

Step-by-Step Solution:

STEP 1: Map Element Positions in A
- Container: Square
- Element 1 (Circle): Left portion of square
- Element 2 (Triangle): Right portion of square
- Spatial arrangement: Circle | Triangle (left | right)

STEP 2: Map Element Positions in B
- Container: Square (same)
- Element 1 (Triangle): Left portion of square
- Element 2 (Circle): Right portion of square
- Spatial arrangement: Triangle | Circle (left | right)

STEP 3: Identify Transformation Type
- Container: Unchanged (square remains square)
- Element types: Unchanged (circle and triangle still present)
- Element positions: Swapped (exchanged positions)
- Transformation: Positional swap/exchange

STEP 4: Identify Elements in C
- Container: Circle
- Element 1 (Square): Left portion of circle
- Element 2 (Line): Right portion of circle
- Current arrangement: Square | Line (left | right)

STEP 5: Apply Position Swap to C
- Element 1 (Square): Moves from left → right
- Element 2 (Line): Moves from right → left
- Result arrangement: Line | Square (left | right)
- Container: Remains circle (unchanged)

Answer: Circle with line on left portion and square on right portion (positions swapped)

Step-by-Step Solution:

STEP 1: Understand Set Operation in A
- Component 1: Circle
- Operation symbol: + (indicates union/combination)
- Component 2: Rectangle
- Components: Overlapping or adjacent

STEP 2: Analyze Result in B
- Single unified region combining both shapes
- Overlapping area counted once (not double-shaded)
- Boundary: Outer perimeter of combined shape
- Operation: UNION (∪) in set theory = Combined area

STEP 3: Identify Components in C
- Component 1: Hexagon
- Operation symbol: + (same as in A)
- Component 2: Trapezoid
- Relationship: Adjacent or overlapping

STEP 4: Construct Union of C
- Merge hexagon and trapezoid
- Create single continuous region
- Outer boundary: Traces perimeter of combined shape
- Interior: All area covered by either polygon

Answer: Single merged region containing all area covered by either hexagon or trapezoid

Step-by-Step Solution:

STEP 1: Identify First Transformation (Rotation)
- Figure A orientation: Triangle pointing right
- Figure B orientation: Triangle pointing upward
- First transformation: 90° anticlockwise rotation

STEP 2: Identify Second Transformation (Scaling)
- Measure A dimensions: Small triangle (base ≈ 20 units)
- Measure B dimensions: Larger triangle (base ≈ 30 units)
- Second transformation: 1.5× scaling (enlargement by 50%)

STEP 3: Determine Transformation Sequence
- Method 1: Rotate first, then scale
- Method 2: Scale first, then rotate
- Result: Same final figure (transformations commute)
- Standard approach: Rotate, then scale

STEP 4: Apply Rotation to C (90° anticlockwise)
- Original C: Vertical rectangle (height > width)
- After rotation: Horizontal rectangle (width > height)
- Internal dot: Rotates with figure to new position

STEP 5: Apply Scaling to Rotated C (1.5×)
- Current dimensions: Already rotated rectangle
- Apply 1.5× to both length and width
- Internal dot: Maintains relative position, scales proportionally

STEP 6: Verify Combined Transformation
- Original: Vertical rectangle, small size, dot at top
- After rotation: Horizontal rectangle, small size, dot on left
- After scaling: Horizontal rectangle, 1.5× larger, dot on left (scaled position)

Answer: Rectangle in horizontal orientation (rotated 90° anticlockwise), dimensions increased by 1.5×, with dot repositioned according to both transformations

Step-by-Step Solution:

STEP 1: Understand Set Operation in A
- Component 1: Circle
- Operation symbol: + (indicates union/combination)
- Component 2: Rectangle
- Components: Overlapping or adjacent

STEP 2: Analyze Result in B
- Single unified region combining both shapes
- Overlapping area counted once (not double-shaded)
- Boundary: Outer perimeter of combined shape
- Operation: UNION (∪) in set theory = Combined area

STEP 3: Identify Components in C
- Component 1: Hexagon
- Operation symbol: + (same as in A)
- Component 2: Trapezoid
- Relationship: Adjacent or overlapping

STEP 4: Construct Union of C
- Merge hexagon and trapezoid
- Create single continuous region
- Outer boundary: Traces perimeter of combined shape
- Interior: All area covered by either polygon

Answer: Single merged region containing all area covered by either hexagon or trapezoid

Step-by-Step Solution:

STEP 1: Identify First Transformation - Rotation
- Figure A: Diamond (square rotated 45°)
- Figure B: Diamond rotated further (additional 45° clockwise)
- First transformation: 45° clockwise rotation

STEP 2: Identify Second Transformation - Scaling
- Measure A diagonal span: Approximately 36 units
- Measure B diagonal span: Approximately 48 units
- Calculate ratio: 48/36 ≈ 1.33 (approximately 1.3)
- Second transformation: 1.3× scaling (30% enlargement)

STEP 3: Identify Third Transformation - Color Inversion
- Figure A fill: White (light), dot: Black (dark)
- Figure B fill: Black (dark), dot: White (light)
- Third transformation: Complete color/shading inversion

STEP 4: Analyze Figure C
- Shape: Rectangle (horizontal orientation)
- Size: Standard (40×34 units)
- Colors: White background, black internal lines
- Internal elements: 2 perpendicular lines (cross pattern)

STEP 5: Apply Transformations in Sequence
- Rotate 45° clockwise → Rectangle becomes tilted (diamond-like)
- Scale 1.3× → All dimensions multiplied by 1.3
- Invert colors → Black background, white lines

Answer: Rectangle rotated 45° clockwise, dimensions enlarged by 1.3×, with inverted coloring (black fill with white cross lines inside)

Step-by-Step Solution:

STEP 1: Identify Numerical Pattern
- A: 1 dot(s)
- B: 3 dot(s)
- Change: +2 dot(s) added

STEP 2: Apply to Figure C
- C: 1 dot(s) (different container)
- Add 2 more dot(s)
- Result: 3 dots total

Answer: Square with 3 dots (2 more than original)

Step-by-Step Solution:

STEP 1: Analyze Figure A
- Outer shape: Circle (can be considered as infinite or 0 sides)
- Inner shape: Diamond (4 sides)

STEP 2: Analyze Figure B
- Outer shape: Hexagon (6 sides)
- Inner shape: Circle (infinite sides)

STEP 3: Identify the Pattern
Looking at the progression:
- A: Outer Circle (∞), Inner Diamond (4)
- B: Outer Hexagon (6), Inner Circle (∞)

The transformation swaps and transforms side counts:
- The inner shape's side count (4) becomes part of the new outer shape
- The pattern shows: 4 → 6 → 8 (increasing by 2 each time)
- Circle (∞) acts as a placeholder that transforms to the next polygon in sequence

STEP 4: Analyze Figure C
- Outer shape: Hexagon (6 sides)
- Inner shape: Diamond (4 sides)

STEP 5: Apply Pattern to C
Following the established pattern:
- The inner diamond has 4 sides → following the 4→6→8 progression, this contributes to the new outer shape
- The outer hexagon has 6 sides → this becomes the new inner shape's side count
- Therefore: New outer = Octagon (8 sides), New inner = Hexagon (6 sides)

STEP 6: Verify Pattern Consistency
- A: Circle (∞) + Diamond (4) → B: Hexagon (6) + Circle (∞)
- C: Hexagon (6) + Diamond (4) → ?: Octagon (8) + Hexagon (6)
- The pattern shows that the inner shape's side count determines the progression of the outer shape

Answer: Octagon (8 sides) with hexagon (6 sides) inside

Key Insight: This problem tests recognition of progressive relationships where shapes evolve based on their side counts, with circle acting as a special case that transforms to the next polygon in the sequence.

Step-by-Step Solution:

STEP 1: Identify First Transformation (Rotation)
- Figure A orientation: Triangle pointing right
- Figure B orientation: Triangle pointing upward
- First transformation: 90° anticlockwise rotation

STEP 2: Identify Second Transformation (Scaling)
- Measure A dimensions: Small triangle (base ≈ 20 units)
- Measure B dimensions: Larger triangle (base ≈ 30 units)
- Second transformation: 1.5× scaling (enlargement by 50%)

STEP 3: Determine Transformation Sequence
- Method 1: Rotate first, then scale
- Method 2: Scale first, then rotate
- Result: Same final figure (transformations commute)
- Standard approach: Rotate, then scale

STEP 4: Apply Rotation to C (90° anticlockwise)
- Original C: Vertical rectangle (height > width)
- After rotation: Horizontal rectangle (width > height)
- Internal dot: Rotates with figure to new position

STEP 5: Apply Scaling to Rotated C (1.5×)
- Current dimensions: Already rotated rectangle
- Apply 1.5× to both length and width
- Internal dot: Maintains relative position, scales proportionally

STEP 6: Verify Combined Transformation
- Original: Vertical rectangle, small size, dot at top
- After rotation: Horizontal rectangle, small size, dot on left
- After scaling: Horizontal rectangle, 1.5× larger, dot on left (scaled position)

Answer: Rectangle in horizontal orientation (rotated 90° anticlockwise), dimensions increased by 1.5×, with dot repositioned according to both transformations

Step-by-Step Solution:

STEP 1: Map Element Positions in A
- Container: Square
- Element 1 (Circle): Left portion of square
- Element 2 (Triangle): Right portion of square
- Spatial arrangement: Circle | Triangle (left | right)

STEP 2: Map Element Positions in B
- Container: Square (same)
- Element 1 (Triangle): Left portion of square
- Element 2 (Circle): Right portion of square
- Spatial arrangement: Triangle | Circle (left | right)

STEP 3: Identify Transformation Type
- Container: Unchanged (square remains square)
- Element types: Unchanged (circle and triangle still present)
- Element positions: Swapped (exchanged positions)
- Transformation: Positional swap/exchange

STEP 4: Identify Elements in C
- Container: Circle
- Element 1 (Square): Left portion of circle
- Element 2 (Line): Right portion of circle
- Current arrangement: Square | Line (left | right)

STEP 5: Apply Position Swap to C
- Element 1 (Square): Moves from left → right
- Element 2 (Line): Moves from right → left
- Result arrangement: Line | Square (left | right)
- Container: Remains circle (unchanged)

Answer: Circle with line on left portion and square on right portion (positions swapped)

Step-by-Step Solution:

STEP 1: Map Element Positions in A
- Container: Square
- Element 1 (Circle): Left portion of square
- Element 2 (Triangle): Right portion of square
- Spatial arrangement: Circle | Triangle (left | right)

STEP 2: Map Element Positions in B
- Container: Square (same)
- Element 1 (Triangle): Left portion of square
- Element 2 (Circle): Right portion of square
- Spatial arrangement: Triangle | Circle (left | right)

STEP 3: Identify Transformation Type
- Container: Unchanged (square remains square)
- Element types: Unchanged (circle and triangle still present)
- Element positions: Swapped (exchanged positions)
- Transformation: Positional swap/exchange

STEP 4: Identify Elements in C
- Container: Circle
- Element 1 (Square): Left portion of circle
- Element 2 (Line): Right portion of circle
- Current arrangement: Square | Line (left | right)

STEP 5: Apply Position Swap to C
- Element 1 (Square): Moves from left → right
- Element 2 (Line): Moves from right → left
- Result arrangement: Line | Square (left | right)
- Container: Remains circle (unchanged)

Answer: Circle with line on left portion and square on right portion (positions swapped)

Step-by-Step Solution:

STEP 1: Identify the Transformation
- Figure A: Triangle in original orientation
- Figure B: Same triangle rotated 90° anticlockwise
- Transformation: 90° anticlockwise rotation

STEP 2: Apply to Figure C
- Figure C: Rectangle (horizontal orientation)
- After 90° anticlockwise rotation: Rectangle becomes vertical
- The shape remains a rectangle, only orientation changes

STEP 3: Verify
- Shape preserved ✓
- Size preserved ✓
- Only orientation changed by 90° anticlockwise ✓

Answer: Rectangle rotated 90° anticlockwise (becomes vertical)

Step-by-Step Solution:

STEP 1: Identify Mirror Axis
- Compare A and B: They are mirror images
- Mirror axis: Horizontal

STEP 2: Properties of horizontal Mirror
- Horizontal mirror: Left ↔ Right swap
- All horizontal positions inverted
- Vertical positions unchanged

STEP 3: Apply to Figure C
- Reflect C across horizontal axis
- Maintain shape type
- Invert orientation

Answer: Figure mirrored across horizontal axis (mirror image of C)

Step-by-Step Solution:

STEP 1: Identify Transformation
- A: Light background, dark internal elements
- B: Dark background, light internal elements
- Rule: Complete color/shading inversion

STEP 2: Apply to Figure C
- C: Light background, dark internal elements
- After inversion: Dark background, light internal elements
- All positions and shapes preserved

Answer: Circle with inverted shading

Step-by-Step Solution:

STEP 1: Count Elements in A and B
- Figure A: Square + 1 dot
- Figure B: Square + 3 dots
- Change: +2 dots added

STEP 2: Apply to Figure C
- Figure C: Different container + 1 dot
- Add 2 more dots
- Result: Container + 3 dots total

Answer: Circle with 3 dots (1 original + 2 added)

Step-by-Step Solution:

STEP 1: Map Element Positions in A
- Container: Square
- Element 1 (Circle): Left portion of square
- Element 2 (Triangle): Right portion of square
- Spatial arrangement: Circle | Triangle (left | right)

STEP 2: Map Element Positions in B
- Container: Square (same)
- Element 1 (Triangle): Left portion of square
- Element 2 (Circle): Right portion of square
- Spatial arrangement: Triangle | Circle (left | right)

STEP 3: Identify Transformation Type
- Container: Unchanged (square remains square)
- Element types: Unchanged (circle and triangle still present)
- Element positions: Swapped (exchanged positions)
- Transformation: Positional swap/exchange

STEP 4: Identify Elements in C
- Container: Circle
- Element 1 (Square): Left portion of circle
- Element 2 (Line): Right portion of circle
- Current arrangement: Square | Line (left | right)

STEP 5: Apply Position Swap to C
- Element 1 (Square): Moves from left → right
- Element 2 (Line): Moves from right → left
- Result arrangement: Line | Square (left | right)
- Container: Remains circle (unchanged)

Answer: Circle with line on left portion and square on right portion (positions swapped)

Step-by-Step Solution:

STEP 1: Map Element Positions in A
- Container: Square
- Element 1 (Circle): Left portion of square
- Element 2 (Triangle): Right portion of square
- Spatial arrangement: Circle | Triangle (left | right)

STEP 2: Map Element Positions in B
- Container: Square (same)
- Element 1 (Triangle): Left portion of square
- Element 2 (Circle): Right portion of square
- Spatial arrangement: Triangle | Circle (left | right)

STEP 3: Identify Transformation Type
- Container: Unchanged (square remains square)
- Element types: Unchanged (circle and triangle still present)
- Element positions: Swapped (exchanged positions)
- Transformation: Positional swap/exchange

STEP 4: Identify Elements in C
- Container: Circle
- Element 1 (Square): Left portion of circle
- Element 2 (Line): Right portion of circle
- Current arrangement: Square | Line (left | right)

STEP 5: Apply Position Swap to C
- Element 1 (Square): Moves from left → right
- Element 2 (Line): Moves from right → left
- Result arrangement: Line | Square (left | right)
- Container: Remains circle (unchanged)

Answer: Circle with line on left portion and square on right portion (positions swapped)

Step-by-Step Solution:

STEP 1: Identify Mirror Axis
- Compare A and B: They are mirror images
- Mirror axis: Horizontal

STEP 2: Properties of horizontal Mirror
- Horizontal mirror: Left ↔ Right swap
- All horizontal positions inverted
- Vertical positions unchanged

STEP 3: Apply to Figure C
- Reflect C across horizontal axis
- Maintain shape type
- Invert orientation

Answer: Figure mirrored across horizontal axis (mirror image of C)

Step-by-Step Solution:

STEP 1: Understand Set Operation in A
- Component 1: Circle
- Operation symbol: + (indicates union/combination)
- Component 2: Rectangle
- Components: Overlapping or adjacent

STEP 2: Analyze Result in B
- Single unified region combining both shapes
- Overlapping area counted once (not double-shaded)
- Boundary: Outer perimeter of combined shape
- Operation: UNION (∪) in set theory = Combined area

STEP 3: Identify Components in C
- Component 1: Hexagon
- Operation symbol: + (same as in A)
- Component 2: Trapezoid
- Relationship: Adjacent or overlapping

STEP 4: Construct Union of C
- Merge hexagon and trapezoid
- Create single continuous region
- Outer boundary: Traces perimeter of combined shape
- Interior: All area covered by either polygon

Answer: Single merged region containing all area covered by either hexagon or trapezoid

Step-by-Step Solution:

STEP 1: Identify First Transformation (Rotation)
- Figure A orientation: Triangle pointing right
- Figure B orientation: Triangle pointing upward
- First transformation: 90° anticlockwise rotation

STEP 2: Identify Second Transformation (Scaling)
- Measure A dimensions: Small triangle (base ≈ 20 units)
- Measure B dimensions: Larger triangle (base ≈ 30 units)
- Second transformation: 1.5× scaling (enlargement by 50%)

STEP 3: Determine Transformation Sequence
- Method 1: Rotate first, then scale
- Method 2: Scale first, then rotate
- Result: Same final figure (transformations commute)
- Standard approach: Rotate, then scale

STEP 4: Apply Rotation to C (90° anticlockwise)
- Original C: Vertical rectangle (height > width)
- After rotation: Horizontal rectangle (width > height)
- Internal dot: Rotates with figure to new position

STEP 5: Apply Scaling to Rotated C (1.5×)
- Current dimensions: Already rotated rectangle
- Apply 1.5× to both length and width
- Internal dot: Maintains relative position, scales proportionally

STEP 6: Verify Combined Transformation
- Original: Vertical rectangle, small size, dot at top
- After rotation: Horizontal rectangle, small size, dot on left
- After scaling: Horizontal rectangle, 1.5× larger, dot on left (scaled position)

Answer: Rectangle in horizontal orientation (rotated 90° anticlockwise), dimensions increased by 1.5×, with dot repositioned according to both transformations

Step-by-Step Solution:

STEP 1: Count Elements in A and B
- Figure A: Square + 1 dot
- Figure B: Square + 2 dots
- Change: +1 dots added

STEP 2: Apply to Figure C
- Figure C: Different container + 1 dot
- Add 1 more dots
- Result: Container + 2 dots total

Answer: Circle with 2 dots (1 original + 1 added)