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    Transformation of shapes Notes for GATE CS

    GATE CS Transformation of shapes: 5 chapters, 19 previous year questions (100% of Spatial Aptitude), 105 practice questions and one solved question from each

    A question from this chapter

    Question 1
    Level 3: Exam Standard

    A square paper is folded in half vertically (right edge over left edge), then in half horizontally (bottom edge over top edge), and then in half vertically again (right edge over left edge). This forms a smaller rectangle.

    Three punches are made on this folded rectangle:

    • Punch 1: Exactly at the bottom-right corner (where all folded edges meet).
    • Punch 2: Exactly at the midpoint of the bottom edge.
    • Punch 3: Strictly in the interior of the rectangle, not touching any edge.

    How many distinct holes will be present when the paper is completely unfolded?

    Question 2
    Level 3: Exam Standard

    Consider the letter 'F' drawn in the Cartesian plane. It undergoes two transformations in sequence:

    1. Reflection across the x-axis.
    2. Rotation by counter-clockwise about the origin.

    Which of the following describes the final orientation of the letter 'F' relative to its original position?

    Question 3
    Level 1: Warm-up

    In a cube net, four squares are arranged in a straight continuous row. If the squares are numbered 1, 2, 3, and 4 from left to right, which pair of squares will become opposite faces when the net is folded into a cube?

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    Transformation of shapes Notes for GATE CS

    GATE CS Transformation of shapes: 5 chapters, 19 previous year questions (100% of Spatial Aptitude), 105 practice questions and one solved question from each chapter.

    About Transformation of shapes Notes

    Full study notes for Transformation of shapes in GATE CS, organised across 5 chapters. Each chapter page explains concepts from the basics with worked examples and the formulas you need.

    Transformation of shapes Weightage in GATE CS

    Transformation of shapes accounts for 19 of 19 Spatial Aptitude previous year questions in our bank (100%), about 1.9 per paper across 10 papers.

    Transformation of shapes Chapter Matrix

    ChapterTopicsPYQsShare of unit PYQsPractice questions
    Paper Folding, Cutting and UnfoldingPaper Folding and Final Dimensions, Cut Patterns and Unfolded Figures316%28
    Symmetry, Rotation and ReflectionLine Symmetry and Symmetric Completion, Rotation and Reflection Transformations316%57
    Shape Assembly, Pattern Completion and 2D VisualizationShape Assembly and Complementary Pieces, Tile Patterns and Missing-Part Completion, Geometric Partitioning and Similar Shapes, Convex and Concave Polygon Identification737%0
    Three-Dimensional Solids, Nets and Orthographic ViewsCube Nets and Solid Formation, Solid Mensuration and Equal-Volume Comparisons, Three-Dimensional Cutting and Subdivision, Surface Curvature and Solid Geometry, Orthographic Views and 3D Reconstruction526%20
    Maps, Directions and Spatial PositioningMap Directions and Relative Positions15%0

    More from Spatial Aptitude

    One Solved Question from Each Transformation of shapes Chapter

    Question 1 · Paper Folding, Cutting and Unfolding MCQ

    A square paper is folded in half vertically (right edge over left edge), then in half horizontally (bottom edge over top edge), and then in half vertically again (right edge over left edge). This forms a smaller rectangle.

    Three punches are made on this folded rectangle:

    • Punch 1: Exactly at the bottom-right corner (where all folded edges meet).
    • Punch 2: Exactly at the midpoint of the bottom edge.
    • Punch 3: Strictly in the interior of the rectangle, not touching any edge.

    How many distinct holes will be present when the paper is completely unfolded?

    1. A.

      11

    2. B.

      13

    3. C.

      15

    4. D.

      17

    Correct Answer:

    B

    Step-by-Step Solution

    Key idea: This is a reverse-engineering question testing layer counting and the effect of fold lines on punch holes. The number of distinct holes depends on how many fold lines the punch intersects.

    Step 1: Trace the folds. Fold 1 (vertical) creates a fold line. Fold 2 (horizontal) creates a fold line. Fold 3 (vertical) creates another fold line. The paper has 8 layers.

    Step 2: Identify the edges of the final rectangle. The bottom-right corner is the intersection of the folded edges. It corresponds to the center of the original square and lies on ALL THREE fold lines.

    Step 3: Analyze Punch 1. It is at the bottom-right corner. Since it lies on all 3 fold lines, unfolding it merges all 8 layers into a single hole at the center of the original square. (1 hole)

    Step 4: Analyze Punch 2. It is at the midpoint of the bottom edge. The bottom edge corresponds to the horizontal fold line only. It does not lie on the vertical fold lines. Thus, it lies on exactly 1 fold line. Unfolding across this fold line merges 2 layers, and the other 2 folds reflect this single hole to 4 locations. (4 holes)

    Step 5: Analyze Punch 3. It is strictly in the interior, lying on 0 fold lines. It pierces 8 layers, and unfolding yields 8 distinct holes. (8 holes)

    Step 6: Total holes = 1 + 4 + 8 = 13.

    Answer: (B)

    Question 2 · Symmetry, Rotation and Reflection MCQ

    Consider the letter 'F' drawn in the Cartesian plane. It undergoes two transformations in sequence:

    1. Reflection across the x-axis.
    2. Rotation by counter-clockwise about the origin.

    Which of the following describes the final orientation of the letter 'F' relative to its original position?

    1. A.

      It is identical to the original 'F'.

    2. B.

      It is a mirror image of the original 'F' across the y-axis.

    3. C.

      It is inverted upside down but not laterally reversed.

    4. D.

      It is rotated by counter-clockwise.

    Correct Answer:

    B

    Step-by-Step Solution

    Key idea: This is a sequential transformation problem. We must track the orientation changes step-by-step using coordinate mapping. Step 1: Analyze Transformation 1 (Reflection across x-axis). Reflection across the x-axis changes to . Crucially, reflection reverses orientation (chirality). Step 2: Analyze Transformation 2 (Rotation by about origin). Rotation by changes to . Apply this to the result of Step 1: The point becomes . Step 3: Identify the Net Transformation. The net mapping is . This transformation corresponds exactly to a reflection across the y-axis. Step 4: Check Orientation. A reflection across the y-axis reverses orientation, which matches our expectation (one reversal from step 1, preservation from step 2 = net reversal). Therefore, the final image is the mirror image of the original across the y-axis.
    Question 3 · Three-Dimensional Solids, Nets and Orthographic Views MCQ

    In a cube net, four squares are arranged in a straight continuous row. If the squares are numbered 1, 2, 3, and 4 from left to right, which pair of squares will become opposite faces when the net is folded into a cube?

    1. A.

      1 and 2

    2. B.

      1 and 3

    3. C.

      2 and 3

    4. D.

      1 and 4

    Correct Answer:

    B

    Step-by-Step Solution

    Key idea: This is a <direct_formula> question testing the opposite face rules in straight rows.

    Step 1: Recall the separation rule for straight rows. In a continuous straight line of squares, any two squares separated by exactly one square will become opposite faces when folded.

    Step 2: Apply the rule to the numbered row (1, 2, 3, 4).

    Step 3: Square 1 and Square 3 are separated by Square 2. Therefore, 1 and 3 are opposite.

    Step 4: Similarly, Square 2 and Square 4 are separated by Square 3. Therefore, 2 and 4 are opposite.

    Step 5: Check the options. Option B correctly identifies the pair 1 and 3.

    Answer: B