Question 5

A square is a shape with four equal sides and four right-angle corners. We can check these properties by counting steps on the grid.

Step 1 — Check Shape A

Let us look at the sides of Shape A. We can count the horizontal and vertical steps between corners. From one corner to the next, we go 2 steps horizontally and 4 steps vertically. For the next side, we go 4 steps horizontally and 2 steps vertically. For the third side, we go 2 steps horizontally and 4 steps vertically. For the fourth side, we go 4 steps horizontally and 2 steps vertically. All sides have the same length. This is because they are all like the diagonal of a rectangle that is 2 units by 4 units.

Now, let us check the corners. Look at two sides that meet at a corner. One side moves 2 steps horizontally and 4 steps vertically. The next side moves 4 steps horizontally and 2 steps vertically. The number of horizontal steps of the first side (2) matches the number of vertical steps of the second side (2). The number of vertical steps of the first side (4) matches the number of horizontal steps of the second side (4). Also, the directions of movement make a perfect corner. This means the corner is a right angle. All four corners of Shape A are right angles. Since all sides are equal and all corners are right angles, Shape A is a square.

Step 2 — Check Shape B

Let us look at the sides of Shape B. From one corner to the next, we go 2 steps horizontally and 3 steps vertically. For the next side, we also go 2 steps horizontally and 3 steps vertically. All sides have the same length. This is because they are all like the diagonal of a rectangle that is 2 units by 3 units.

Now, let us check the corners. Look at two sides that meet at a corner. One side moves 2 steps horizontally and 3 steps vertically. The next side also moves 2 steps horizontally and 3 steps vertically. For a right angle, these steps should swap. For example, if one side is (2,3), the next side should be (3,2) with one direction changed. Since the steps do not swap, the corners are not right angles. So, Shape B is not a square.

Step 3 — Check Shape C

Let us look at the sides of Shape C. From one corner to the next, we go 6 steps horizontally and 1 step vertically. For the next side, we go 2 steps horizontally and 4 steps vertically. The horizontal steps are different (6 and 2). The vertical steps are different (1 and 4). This means the lengths of these two sides are not equal. Since not all sides are equal, Shape C is not a square.

Step 4 — Check Shape D

Let us look at the sides of Shape D. From one corner to the next, we go 1 step horizontally and 2 steps vertically. For the next side, we go 2 steps horizontally and 1 step vertically. For the third side, we go 1 step horizontally and 2 steps vertically. For the fourth side, we go 2 steps horizontally and 1 step vertically. All sides have the same length. This is because they are all like the diagonal of a rectangle that is 1 unit by 2 units.

Now, let us check the corners. Look at two sides that meet at a corner. One side moves 1 step horizontally and 2 steps vertically. The next side moves 2 steps horizontally and 1 step vertically. The number of horizontal steps of the first side (1) matches the number of vertical steps of the second side (1). The number of vertical steps of the first side (2) matches the number of horizontal steps of the second side (2). Also, the directions of movement make a perfect corner. This means the corner is a right angle. All four corners of Shape D are right angles. Since all sides are equal and all corners are right angles, Shape D is a square.

Answer

(i) Shapes A and D are squares.