Constructions and Tilings | FIO

Question 23

How can the tangram pieces be rearranged to form each of the following figures?

Question diagram 1
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Solution
Understand the Question
  • A standard tangram puzzle consists of 7 geometric pieces:
    • Two large right triangles: LT1\text{LT1} and LT2\text{LT2}
    • One medium right triangle: MT\text{MT}
    • Two small right triangles: ST1\text{ST1} and ST2\text{ST2}
    • One square: SQ\text{SQ}
    • One parallelogram: PG\text{PG}
  • All seven pieces must be used for each figure without overlapping.

Diagram 1

(i) Letter C

Answer

(i) * LT1: Forms the bottom horizontal bar (right angle points bottom-left).

  • LT2: Forms the top horizontal bar (right angle points top-left).
  • SQ: Placed vertically in the middle, connecting LT1 and LT2.
  • MT: Placed above the Square, forming the upper inner curve.
  • ST1: Placed below the Square, forming the lower inner curve.
  • PG: Placed to the right of MT, forming the outer top arm.
  • ST2: Placed to the right of ST1, forming the outer bottom arm.

(ii) Pinwheel

Answer

(ii) * The central square void is left empty.

  • LT1 & ST1: Complete the top-right blade (right angles point towards the center).
  • LT2 & ST2: Complete the bottom-left blade (right angles point towards the center).
  • MT: Forms the bottom-right blade (right angle points towards the center).
  • PG: Forms the top-left blade (acute angle points towards the center).

(iii) Cat

Answer

(iii) * LT1: Forms the lower body (right angle points down-left).

  • LT2: Forms the upper body (right angle points up-left).
  • ST1: Forms the head (on top-left of LT2).
  • ST2: Forms the ear (on top-right of ST1).
  • MT: Forms the front leg (below the front of LT1).
  • SQ: Forms the back leg (below the back of LT1).
  • PG: Forms the tail (at the back of LT1, pointing up).

(iv) Duck/Bird

Answer

(iv) * LT1: Forms the lower body (right angle points down-left).

  • LT2: Forms the upper body/wing (right angle points up-left).
  • MT: Forms the head (at the front of LT2).
  • ST1: Forms the beak (at the front of MT).
  • PG: Forms the tail (at the back of LT1, pointing up).
  • SQ: Forms the upper leg (below LT1).
  • ST2: Forms the foot (below the Square).

(v) Running Man

Answer

(v) * LT1: Forms the torso (right angle points down-left).

  • LT2: Forms the upper body (right angle points up-left).
  • SQ: Forms the head (on top of LT2).
  • MT: Forms an arm (extends from LT2, pointing up-right).
  • PG: Forms the front leg (extends from LT1, pointing forward).
  • ST1: Forms the back leg (extends from LT1, pointing down-left).
  • ST2: Forms the foot (at the end of ST1, pointing down-left).

(vi) Letter M

Answer

(vi) * LT1: Forms the left outer leg (right angle points down-left).

  • LT2: Forms the right outer leg (right angle points down-right).
  • SQ: Forms the top middle (between LT1 and LT2).
  • MT: Forms the left inner slant (below the left of SQ).
  • PG: Forms the right inner slant (below the right of SQ).
  • ST1: Forms the bottom-left point (below MT).
  • ST2: Forms the bottom-right point (below PG).

(vii) Bat/Monster

Answer

(vii) * LT1: Forms the top-left point (right angle points up-left).

  • LT2: Forms the top-right point (right angle points up-right).
  • SQ: Forms the upper body (between LT1 and LT2).
  • MT: Forms the middle body (below the Square).
  • ST1: Forms the bottom-left jaw/leg (below the left of MT).
  • ST2: Forms the bottom-right jaw/leg (below the right of MT).
  • PG: Forms the bottom jaw/tail (connects ST1 and ST2).

(viii) Whale

Answer

(viii) * LT1: Forms the main body (right angle points down-left).

  • LT2: Forms the upper body (right angle points up-left).
  • MT: Forms the upper tail fin (at the back of LT2).
  • PG: Forms the lower tail fin (at the back of LT1).
  • SQ: Forms the head (at the front of LT1).
  • ST1: Forms the upper fin (on top of the Square).
  • ST2: Forms the lower fin (below the Square).

(ix) Fish

Answer

(ix) * LT1: Forms the main body (right angle points down-left).

  • LT2: Forms the upper body (right angle points up-left).
  • MT: Forms the head (at the front of LT1).
  • PG: Forms the upper tail fin (at the back of LT2).
  • ST1: Forms the lower tail fin (at the back of LT1).
  • SQ: Forms the upper fin (on top of LT2).
  • ST2: Forms the lower fin (below LT1).

(x) Face

Answer

(x) * LT1: Forms the back of the head (right angle points down-left).

  • LT2: Forms the top of the head (right angle points up-left).
  • SQ: Forms the forehead/eye (at the front of LT2).
  • MT: Forms the nose (below the Square).
  • PG: Forms the mouth (below the Medium Triangle).
  • ST1: Forms the chin (below the Parallelogram).
  • ST2: Forms the neck (below LT1).
Common Mistakes
  • Overlapping Pieces: Tangram pieces must touch edge-to-edge or vertex-to-vertex, but should never overlap each other.
  • Missing Pieces: Every tangram figure requires all 77 pieces to be used without omitting any.
  • Parallelogram Chirality: The parallelogram is chiral (has no reflective symmetry in the plane); you may need to flip it over to match the required orientation.

More questions in FIO

Q1

When constructing the perpendicular bisector, is it necessary to have the same radius for the arcs above and below XYXY? Explore this through construction, and then justify your answer.

[Hint 1: Any point that is of the same distance from XX and YY lies on the perpendicular bisector.

Hint 2: We can draw the whole line if any two of its points are known.]

Q2

Is it necessary to construct the pairs of arcs above and below XYXY? Instead, can we construct both the pairs of arcs on the same side of XYXY? Explore this through construction, and then justify your answer.

Q3

While constructing one pair of intersecting arcs, is it necessary that we use the same radii for both of them ? Explore this through construction, and then justify your answer.

Q4

Recreate this design using only a ruler and compass —

Q5

Justify why AB\text{AB} in Fig. 6.4 is the perpendicular bisector.

Q6

Can you think of different methods to construct a 90∘90^\circ angle at a given point on a line using a rope?

Q7

Construct at least 4 different angles. Draw their bisectors.

Q8

Construct the 8-petalled figure shown in Fig. 6.5.

Q9

In Step 2 of angle bisection, if arcs of equal radius are drawn on the other side, as shown in the figure, will the line OCOC still be an angle bisector? Explore this through construction, and then justify your answer.

Q10

What are the other angles that can be constructed using angle bisection? Can you construct 65.5∘65.5^\circ angle?

Q11

Come up with a method to construct the angle bisector using a rope.

Q12

Construct the following figure.

How do we construct the petals so that they are of the maximum possible size within a given square?

Q13

Construct at least 4 different angles in different orientations without taking any measurement. Make a copy of all these angles.

Q14

Construct the Fig. 6.6.

Q15

Construct 4 pairs of parallel lines in different orientations.

Q16

Construct the following figure.

Q17

Use support lines in Fig. 6.11 to construct a pointed arch. Make different arches, by changing the radius of the arcs.

Q18

Make your own arch designs.

Q19

Construct the following figures:

Q20

Optical Illusion: Do you notice anything interesting about the following figure? How does this happen? Recreate this in your notebook.

Q21

Construct this figure.

[Hint: Find the angles in this figure.]

Q22

Draw a line ll and mark a point PP anywhere outside the line. Construct a perpendicular to the given line ll through PP.

[Hint: Find a line segment on ll whose perpendicular bisector passes through PP.]

Q23

How can the tangram pieces be rearranged to form each of the following figures?

Q24

Are the following tilings possible?

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