Question 1

A square piece of paper is folded once along its diagonal. After unfolding, how many triangular regions are created by the fold line?

Accepted Answer

When a square is folded along its diagonal, the fold line runs from one corner to the opposite corner. Upon unfolding, this single fold line divides the square into exactly 2 triangular regions of equal size. The diagonal of a square always creates two congruent right triangles.

Question 2

A square piece of paper is folded along its diagonal, then the resulting triangle is folded in half along a line parallel to its base. After unfolding completely, how many distinct regions are created on the original square?

Accepted Answer

Setup: This question tests the ability to mentally track fold lines and visualize their imprint on the original paper after complete unfolding.

Solution:
1. Start with a square ABCD. Fold along diagonal AC, bringing triangle ABC onto triangle ACD.
2. The resulting shape is a right isosceles triangle with the right angle at C.
3. Now fold this triangle in half along a line parallel to the base (hypotenuse AC). This line runs from the midpoint of one leg to the midpoint of the other leg, creating a smaller triangle and a trapezoid.
4. When you unfold the first fold (the diagonal), the fold line from step 3 creates a corresponding crease on the other half of the square.
5. The diagonal fold creates 1 line. The second fold, when unfolded through the diagonal, creates 2 additional lines (one on each half of the square, symmetrically placed).
6. These 3 lines intersect the square creating distinct regions. The diagonal divides the square into 2 regions. The two parallel-to-diagonal lines (from the second fold) each cross both triangular halves, adding 4 more regions.
7. Total: 2 + 4 = 6 distinct regions.

Trap Explanation: Option B (8 regions) is the most tempting because students often count each fold as creating 2 new regions independently, leading to 2 + 2 + 2 = 6, then mistakenly add the intersections again. Option C (4 regions) results from only counting the diagonal fold's effect and ignoring the second fold's contribution entirely. Option D (10 regions) comes from overcounting intersections by treating each crease as if it independently divides all existing regions.

Shortcut: For paper folding problems, use the rule: if you make n non-overlapping folds on a flat paper, the maximum number of regions created is 2^n. However, when folds are dependent (one fold affects where another can go), manually trace the crease pattern. Here: 1 diagonal + 2 symmetric creases from the parallel fold = 3 total crease lines. A square crossed by 3 lines in this configuration (1 diagonal, 2 symmetric to it) creates 6 regions. Verify by sketching.

Question 3

A square piece of paper is folded diagonally once, then the resulting triangle is folded in half along its altitude from the right angle. After unfolding completely, how many distinct regions are created on the original square?

Accepted Answer

Setup: A square ABCD is folded along diagonal AC, creating a triangle. The folded triangle is then folded along the altitude from the right angle (vertex A or C, depending on orientation) to the hypotenuse. When unfolded, the crease pattern must be traced carefully.

Solution:
1. Start with square ABCD. Fold along diagonal AC → creates a double-layer triangle.
2. The resulting triangle has vertices at A, C, and one corner (say B). The right angle is at B.
3. Fold this triangle along the altitude from B to AC. This altitude bisects AC and is perpendicular to it.
4. When folded, this creates one additional crease perpendicular to AC, passing through its midpoint.
5. Upon complete unfolding:
   - The first fold (diagonal AC) creates 1 crease dividing the square into 2 regions.
   - The second fold (altitude from right angle) creates 1 crease perpendicular to AC through its midpoint.
   - These two creases intersect at one point (the midpoint of AC).
   - The altitude crease extends from one edge of the square to the opposite edge, passing through the interior.
6. Total creases: 2 (one diagonal, one perpendicular bisector of the diagonal).
7. Two non-parallel lines intersecting inside a square create 4 regions. The perpendicular bisector of the diagonal further subdivides these, creating 8 distinct regions total.

Therefore, the answer is B (8 regions).

RRB Group D Paper Folding & Cutting

Paper Folding & Cutting— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Paper Folding & Cutting — Practice Questions

60-Second Revision — Paper Folding & Cutting