Playing with Constructions | FIO

Question 6

Think: Is it possible to reason out if the sides are equal or not, and if the angles are right or not without using any measuring instruments in the above figure? Can we do this by only looking at the position of corners in the dot grid?

Question diagram 1
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Solution
Understand the Question
  • Yes, it is possible to determine whether sides are equal and angles are 90∘90^\circ without measuring instruments by counting the horizontal and vertical grid steps between corner points on the dot grid.
  • Side Lengths: If two line segments span the same horizontal and vertical grid distances, their lengths are equal (by the Pythagorean theorem).
  • Right Angles (90∘90^\circ): Two lines are perpendicular (meet at 90∘90^\circ) if the product of their slopes is −1-1, or if a horizontal/vertical shift of (x,y)(x, y) on one side corresponds to an orthogonal shift of (−y,x)(-y, x) or (y,−x)(y, -x) on the adjacent side.

Think: Is it possible to reason out if the sides are equal or not, and if the angles are right or not without using any measuring instruments in the above figure? Can we do this by only looking at the position of corners in the dot grid?

Step 1 · Check Shape A

Count the grid steps between consecutive corners:

  • Top corner to right corner: 33 steps right, 33 steps down
  • Right corner to bottom corner: 33 steps left, 33 steps down
  • Bottom corner to left corner: 33 steps left, 33 steps up
  • Left corner to top corner: 33 steps right, 33 steps upDiagram 1

Since each side spans 33 horizontal and 33 vertical units, all 44 sides are equal in length.

Check the angles using slope (steepness): Slope of top-right side=−33=−1\text{Slope of top-right side} = \dfrac{-3}{3} = -1 Slope of top-left side=−3−3=1\text{Slope of top-left side} = \dfrac{-3}{-3} = 1

Product of slopes=(−1)×(1)=−1\text{Product of slopes} = (-1) \times (1) = -1

Since the product of slopes is −1-1, the sides are perpendicular (90∘90^\circ). All four corners form 90∘90^\circ angles.

Shape A is a square.\text{Shape A is a square.}

Step 2 · Check Shape B

Count the grid steps between consecutive corners:

  • Top corner to right corner: 33 steps right, 33 steps down
  • Right corner to bottom corner: 33 steps left, 33 steps down
  • Bottom corner to left corner: 33 steps left, 33 steps up
  • Left corner to top corner: 33 steps right, 33 steps upDiagram 2

All sides have identical step spans (33 units horizontal, 33 units vertical), so all sides are equal.

Checking slopes for perpendicularity: Slope1=−33=−1,Slope2=−3−3=1\text{Slope}_1 = \dfrac{-3}{3} = -1, \quad \text{Slope}_2 = \dfrac{-3}{-3} = 1

Product=(−1)×1=−1\text{Product} = (-1) \times 1 = -1

All four angles are 90∘90^\circ.

Shape B is a square.\text{Shape B is a square.}

Step 3 · Check Shape C

Count the grid steps between consecutive corners:

  • Top-left to top-right: 55 steps right, 00 steps vertical
  • Top-right to bottom-right: 33 steps right, 66 steps down
  • Bottom-right to bottom-left: 55 steps left, 00 steps vertical
  • Bottom-left to top-left: 33 steps left, 66 steps upDiagram 3

The side lengths are not all equal (55 units vs. 32+62=45\sqrt{3^2 + 6^2} = \sqrt{45} units).

Since the top side is horizontal and the adjacent side is slanted, the angles are not 90∘90^\circ.

Shape C is not a square.\text{Shape C is not a square.}

Step 4 · Check Shape D

Count the grid steps between consecutive corners:

  • Top-left to top-right: 22 steps right, 11 step up
  • Top-right to bottom-right: 22 steps right, 44 steps down
  • Bottom-right to bottom-left: 22 steps left, 11 step down
  • Bottom-left to top-left: 22 steps left, 44 steps upDiagram 4

The adjacent sides have different step spans ({2,1}\{2, 1\} vs {2,4}\{2, 4\}), so all sides are not equal.

Check the angles: Slope of top side=12\text{Slope of top side} = \dfrac{1}{2} Slope of left side=4−2=−2\text{Slope of left side} = \dfrac{4}{-2} = -2

Product of slopes=(12)×(−2)=−1\text{Product of slopes} = \left(\dfrac{1}{2}\right) \times (-2) = -1

All angles are 90∘90^\circ, but because adjacent sides are unequal, Shape D is a rectangle, not a square.

Shape D is not a square.\text{Shape D is not a square.}

Answer

Yes, it is possible to determine side equality and right angles by counting grid steps. A shape is a square only if all sides are equal and all angles are 90∘90^\circ.

(i) Which shape(s) are squares based on the dot grid analysis?

Answer

(i) Shape A (and Shape B) are squares because all sides are equal and all angles are 90∘90^\circ.

Common Mistakes
  • Assuming Visual Right Angles: Assuming an angle is 90∘90^\circ by appearance alone rather than verifying the step components or checking that the product of slopes equals −1-1.
  • Confusing Rectangles with Squares: Identifying Shape D as a square because all its angles are 90∘90^\circ, while overlooking that its adjacent sides are unequal in length.

More questions in FIO

Q1

What radius should be taken in the compass to get this half circle? What should be the length of AXAX?

Q2

Take a central line of a different length and try to draw the wave on it.

Q3

Try to recreate the figure where the waves are smaller than a half circle (as appearing in the neck of the figure, 'A Person'). The challenge here is to get both the waves to be identical. This may be tricky!

Q4

Draw the rectangle and four squares configuration (shown in Fig. 8.3) on a dot paper.

What did you do to recreate this figure so that the four squares are placed symmetrically around the rectangle? Discuss with your classmates.

Q5

Identify if there are any squares in this collection. Use measurements if needed.

Q6

Think: Is it possible to reason out if the sides are equal or not, and if the angles are right or not without using any measuring instruments in the above figure? Can we do this by only looking at the position of corners in the dot grid?

Q7

Draw at least 3 rotated squares and rectangles on a dot grid. Draw them such that their corners are on the dots. Verify if the squares and rectangles that you have drawn satisfy their respective properties.

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