Question 5
Identify if there are any squares in this collection. Use measurements if needed.
A polygon is a square if and only if:
- All four sides are equal in length.
- All four interior angles are right angles ().
On a square grid:
- Side length: Calculated using the horizontal and vertical steps between consecutive corners.
- Right angles: Adjacent sides are perpendicular if the horizontal and vertical steps swap magnitudes (e.g., a step of followed by a perpendicular step of ).
Step 1 · Check Shape A
- Side lengths: Moving from one corner to the next gives grid steps of units and units for all four sides. Since each side forms the hypotenuse of a right triangle with legs and , all four sides are equal.
- Angles: Adjacent sides swap steps from to in perpendicular directions, making every corner a right angle ().
Therefore, Shape A is a square.
Step 2 · Check Shape B
- Side lengths: Each side spans horizontal steps and vertical steps, so all four sides are equal.
- Angles: Adjacent sides do not swap step directions (both use without the perpendicular swap to ), so the corners are not right angles.
Therefore, Shape B is not a square (it is a rhombus).
Step 3 · Check Shape C
- Side lengths: One side spans horizontal steps and vertical step, while the adjacent side spans horizontal steps and vertical steps.
- Since the side lengths are unequal, the shape cannot be a square.
Therefore, Shape C is not a square.
Step 4 · Check Shape D
- Side lengths: All four sides span grid steps of unit and units, so all four sides are equal.
- Angles: Adjacent sides alternate steps between and in perpendicular directions, forming angles at all four corners.
Therefore, Shape D is a square.
Shapes A and D are squares.
- Confusing a Rhombus with a Square: Shape B has four equal sides, but its angles are not , making it a rhombus rather than a square.
- Judging by Tilt: Assuming tilted shapes (like A and D) are not squares just because they are rotated on the grid.
More questions in FIO
What radius should be taken in the compass to get this half circle? What should be the length of ?
Take a central line of a different length and try to draw the wave on it.
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!
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.
Identify if there are any squares in this collection. Use measurements if needed.
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?
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.