Playing with Constructions | FIO

Question 7

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.

Solution

Understand the Question

A square requires all $4$ sides to be equal in length and all $4$ interior angles to be $90^\circ$ (right angles).
A rectangle requires opposite sides to be equal in length and all $4$ interior angles to be $90^\circ$ .
On a dot grid, side lengths can be compared using horizontal ( $h$ ) and vertical ( $v$ ) step patterns between vertices.
Two perpendicular line segments meeting at a corner satisfy the condition: $(h_1 \times h_2) + (v_1 \times v_2) = 0$

Step 1 · Define Coordinates of the Three Shapes

Diagram 1

We plot three rotated figures with vertices on the grid dots:

Shape 1 (Square $ABCD$ ): $A(1, 3)$ , $B(3, 4)$ , $C(4, 2)$ , $D(2, 1)$
Shape 2 (Rectangle $EFGH$ ): $E(2, 2)$ , $F(5, 3)$ , $G(3, 9)$ , $H(0, 8)$
Shape 3 (Square $IJKL$ ): $I(6, 1)$ , $J(8, 2)$ , $K(7, 4)$ , $L(5, 3)$

Step 2 · Verify Properties of Square 1 ( $ABCD$ )

Side Lengths (Step Patterns):

Side $AB$ : $2$ steps right, $1$ step up
Side $BC$ : $1$ step right, $2$ steps down
Side $CD$ : $2$ steps left, $1$ step down
Side $DA$ : $1$ step left, $2$ steps up

All sides follow the $(2, 1)$ step pattern, so all four side lengths are equal.

Right Angle Checks:

For $\angle A$ ( $AD$ with $h_1 = -1, v_1 = 2$ and $AB$ with $h_2 = 2, v_2 = 1$ ):

\begin{aligned} (-1 \times 2) + (2 \times 1) &= -2 + 2 \\ &= 0 \end{aligned}

For $\angle B$ ( $BA$ with $h_1 = -2, v_1 = -1$ and $BC$ with $h_2 = 1, v_2 = -2$ ):

\begin{aligned} (-2 \times 1) + (-1 \times -2) &= -2 + 2 \\ &= 0 \end{aligned}

Similarly, $\angle C = 90^\circ$ and $\angle D = 90^\circ$ .

Therefore, Shape 1 is a square.

Step 3 · Verify Properties of Rectangle 1 ( $EFGH$ )

Side Lengths (Step Patterns):

Side $EF$ : $3$ steps right, $1$ step up
Side $FG$ : $2$ steps left, $6$ steps up
Side $GH$ : $3$ steps left, $1$ step down
Side $HE$ : $2$ steps right, $6$ steps down

Opposite sides $EF = GH$ (step pattern $3, 1$ ) and $FG = HE$ (step pattern $2, 6$ ) are equal.

Right Angle Checks:

For $\angle E$ ( $EH$ with $h_1 = 2, v_1 = -6$ and $EF$ with $h_2 = 3, v_2 = 1$ ):

\begin{aligned} (2 \times 3) + (-6 \times 1) &= 6 - 6 \\ &= 0 \end{aligned}

For $\angle F$ ( $FE$ with $h_1 = -3, v_1 = -1$ and $FG$ with $h_2 = -2, v_2 = 6$ ):

\begin{aligned} (-3 \times -2) + (-1 \times 6) &= 6 - 6 \\ &= 0 \end{aligned}

Similarly, $\angle G = 90^\circ$ and $\angle H = 90^\circ$ .

Therefore, Shape 2 is a rectangle.

Step 4 · Verify Properties of Square 2 ( $IJKL$ )

Side Lengths (Step Patterns):

Side $IJ$ : $2$ steps right, $1$ step up
Side $JK$ : $1$ step left, $2$ steps up
Side $KL$ : $2$ steps left, $1$ step down
Side $LI$ : $1$ step right, $2$ steps down

All sides follow the $(2, 1)$ step pattern, so all four side lengths are equal.

Right Angle Checks:

For $\angle I$ ( $LI$ with $h_1 = 1, v_1 = -2$ and $IJ$ with $h_2 = 2, v_2 = 1$ ):

\begin{aligned} (1 \times 2) + (-2 \times 1) &= 2 - 2 \\ &= 0 \end{aligned}

For $\angle J$ ( $IJ$ with $h_1 = -2, v_1 = -1$ and $JK$ with $h_2 = -1, v_2 = 2$ ):

\begin{aligned} (-2 \times -1) + (-1 \times 2) &= 2 - 2 \\ &= 0 \end{aligned}

Similarly, $\angle K = 90^\circ$ and $\angle L = 90^\circ$ .

Therefore, Shape 3 is a square.

Answer

The three shapes drawn on the dot grid are verified as follows:

Square 1 ( $ABCD$ ): All $4$ sides are equal and all $4$ angles are $90^\circ$ .
Rectangle 1 ( $EFGH$ ): Opposite sides are equal and all $4$ angles are $90^\circ$ .
Square 2 ( $IJKL$ ): All $4$ sides are equal and all $4$ angles are $90^\circ$ .

Common Mistakes

Counting Dots Instead of Steps: Counting the number of dots rather than the horizontal and vertical intervals between dots leads to incorrect side lengths.
Ignoring Right Angles: Assuming a 4-sided figure with equal sides is automatically a square without verifying that adjacent sides are perpendicular ( $90^\circ$ angles).

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