Quadrilaterals | A

Question 8

Which Quad?

Gameplay

  1. Fold a sheet into half.
  2. Now, fold it once more into a quarter.
  3. Make a triangular crease at the corner that is at the middle of the paper.
  4. Open the sheet. What is the shape formed by the creases?
  5. How would you fold the quarter paper to get the kinds of creases shown in the following image.
  6. How would you fold the quarter paper such that a square is formed?
Question diagram 1
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Solution
Understand the Question
  • Folding a rectangular or square sheet in half and then into quarters divides the paper into four congruent quadrants that meet at the paper's center point.
  • Creasing a corner at this central point creates a symmetrical fold across all four quadrants when unfolded.
  • The resulting geometric figure depends on the symmetry of the folds and the shape of the paper.

(i) Open the sheet. What is the shape formed by the creases?

Step 1 · Identify the Shape Formed by the Creases

Folding the paper into quarters and making a single straight triangular crease across the central folded corner creates four equal crease lines when unfolded.Diagram 1

Since all four sides are equal in length, the quadrilateral formed by the creases is a rhombus.

Answer

(i) Rhombus

(ii) How would you fold the quarter paper to get the kinds of creases shown in the following image.

Step 1 · Create Nested Concentric Creases

To get nested diamond-shaped creases, repeatedly make parallel triangular creases at the innermost corner of the quarter-folded paper at different distances from the center before unfolding.

Answer

(ii) Repeatedly make parallel triangular creases at the innermost corner of the folded quarter paper.

(iii) How would you fold the quarter paper such that a square is formed?

Step 1 · Condition for Forming a Square

A square is a rhombus whose interior angles are all 90∘90^\circ.

To form a square:

  • Start with a square sheet of paper so that the quarter fold has perpendicular lines of symmetry.
  • Fold the inner central corner such that the crease makes an angle of 45∘45^\circ with both edges (an isosceles right triangle fold).
Answer

(iii) Start with a square paper and fold the central corner equally along both edges at an angle of 45∘45^\circ.

Common Mistakes
  • Wrong Corner Folded: Folding one of the open outer corners instead of the central closed corner will create four separate corner creases rather than a single centered quadrilateral.
  • Assuming Rhombus is Always a Square: A straight cut/crease across the folded corner produces a rhombus in general. It only becomes a square if the original sheet is square and the fold is made at 45∘45^\circ to both perpendicular edges.

More questions in A

Q1

Are squares the only quadrilaterals that have equal sidelengths? Let us explore this question through construction.

Draw two equal sides ADAD and ABAB, that are not perpendicular to each other.

Q2

Can we complete this quadrilateral so that all its sides are of the same length?

Mark a point C whose distance from B and D is equal to AB\text{AB} (or AD\text{AD}). To do this, measure AB\text{AB} using a compass. Keeping this length as the radius, cut arcs from B and D.

Q3

Place two rubber bands perpendicular to each other, forming diagonals of equal length. Join the ends.

What is the quadrilateral that you get? Justify your answer.

Q4

Extend one of the diagonals on both sides by 2 cm2\text{ cm}.

What quadrilateral will you get now? Justify your answer.

Q5

Take two cardboard cutouts of an equilateral triangle of sidelength 8 cm8\text{ cm}.

  • Can you join them to get a quadrilateral?
  • What type of a quadrilateral is this? Justify your answer.
Q6

Take two cardboard cutouts of an isosceles triangle with sidelengths 8 cm8\text{ cm}, 8 cm8\text{ cm}, and 6 cm6\text{ cm}.

  • What are the different ways they can be joined to get a quadrilateral?
  • What quadrilaterals are these? Justify your answers.
Q7

Take two cardboard cutouts of a scalene triangle with sides 6 cm6\text{ cm}, 9 cm9\text{ cm}, and 12 cm12\text{ cm}.

  • What are the different ways they can be joined to get a quadrilateral?
  • Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral.
Q8

Which Quad?

Gameplay

  1. Fold a sheet into half.
  2. Now, fold it once more into a quarter.
  3. Make a triangular crease at the corner that is at the middle of the paper.
  4. Open the sheet. What is the shape formed by the creases?
  5. How would you fold the quarter paper to get the kinds of creases shown in the following image.
  6. How would you fold the quarter paper such that a square is formed?
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