Question 7

We can draw rotated squares and rectangles on a dot grid and check their properties.

Step 1 — Drawing the shapes

Let us draw three shapes on a dot grid. We will use dots as corners for each shape.

Shape 1 is a square with corners at: A = (1, 3) B = (3, 4) C = (4, 2) D = (2, 1)

Shape 2 is a rectangle with corners at: E = (2, 2) F = (5, 3) G = (3, 9) H = (0, 8)

Shape 3 is another square with corners at: I = (6, 1) J = (8, 2) K = (7, 4) L = (5, 3)

Step 2 — Verifying Square 1

Let us check if Shape 1 (ABCD) is a square. A square has four equal sides and four right angles.

First, we check the side lengths by counting steps between dots. Side AB: From A(1,3) to B(3,4), we go 2 steps right and 1 step up. Side BC: From B(3,4) to C(4,2), we go 1 step right and 2 steps down. Side CD: From C(4,2) to D(2,1), we go 2 steps left and 1 step down. Side DA: From D(2,1) to A(1,3), we go 1 step left and 2 steps up.

All sides have the same "step pattern". They are either (2 horizontal, 1 vertical) or (1 horizontal, 2 vertical). So, all four sides are equal in length.

Next, we check the angles. We look at two sides meeting at a corner. For side 1: horizontal steps = $h_1$, vertical steps = $v_1$. For side 2: horizontal steps = $h_2$, vertical steps = $v_2$. If $(h_1 \times h_2) + (v_1 \times v_2)$ equals 0, then the angle is a right angle.

Let us check angle A: Side AD: $h_1 = -1$ (left 1), $v_1 = 2$ (up 2). Side AB: $h_2 = 2$ (right 2), $v_2 = 1$ (up 1). $(-1 \times 2) + (2 \times 1)$ $= -2 + 2$ $= 0$ So, angle A is a right angle.

Let us check angle B: Side BA: $h_1 = -2$ (left 2), $v_1 = -1$ (down 1). Side BC: $h_2 = 1$ (right 1), $v_2 = -2$ (down 2). $(-2 \times 1) + (-1 \times -2)$ $= -2 + 2$ $= 0$ So, angle B is a right angle.

We can check angles C and D in the same way. They will also be right angles.

$\boxed{\text{Shape 1 is a square.}}$

Step 3 — Verifying Rectangle 1

Let us check if Shape 2 (EFGH) is a rectangle. A rectangle has opposite sides equal and four right angles.

First, we check the side lengths by counting steps. Side EF: From E(2,2) to F(5,3), we go 3 steps right and 1 step up. Side FG: From F(5,3) to G(3,9), we go 2 steps left and 6 steps up. Side GH: From G(3,9) to H(0,8), we go 3 steps left and 1 step down. Side HE: From H(0,8) to E(2,2), we go 2 steps right and 6 steps down.

Side EF (right 3, up 1) and Side GH (left 3, down 1) have the same step pattern. So, they are equal. Side FG (left 2, up 6) and Side HE (right 2, down 6) have the same step pattern. So, they are equal. Opposite sides are equal in length.

Next, we check the angles using the same method as before.

Let us check angle E: Side EH: $h_1 = 2$ (right 2), $v_1 = -6$ (down 6). Side EF: $h_2 = 3$ (right 3), $v_2 = 1$ (up 1). $(2 \times 3) + (-6 \times 1)$ $= 6 - 6$ $= 0$ So, angle E is a right angle.

Let us check angle F: Side FE: $h_1 = -3$ (left 3), $v_1 = -1$ (down 1). Side FG: $h_2 = -2$ (left 2), $v_2 = 6$ (up 6). $(-3 \times -2) + (-1 \times 6)$ $= 6 - 6$ $= 0$ So, angle F is a right angle.

We can check angles G and H in the same way. They will also be right angles.

$\boxed{\text{Shape 2 is a rectangle.}}$

Step 4 — Verifying Square 2

Let us check if Shape 3 (IJKL) is a square.

First, we check the side lengths by counting steps. Side IJ: From I(6,1) to J(8,2), we go 2 steps right and 1 step up. Side JK: From J(8,2) to K(7,4), we go 1 step left and 2 steps up. Side KL: From K(7,4) to L(5,3), we go 2 steps left and 1 step down. Side LI: From L(5,3) to I(6,1), we go 1 step right and 2 steps down.

All sides have the same "step pattern". They are either (2 horizontal, 1 vertical) or (1 horizontal, 2 vertical). So, all four sides are equal in length.

Next, we check the angles.

Let us check angle I: Side LI: $h_1 = 1$ (right 1), $v_1 = -2$ (down 2). Side IJ: $h_2 = 2$ (right 2), $v_2 = 1$ (up 1). $(1 \times 2) + (-2 \times 1)$ $= 2 - 2$ $= 0$ So, angle I is a right angle.

Let us check angle J: Side IJ: $h_1 = -2$ (left 2), $v_1 = -1$ (down 1). Side JK: $h_2 = -1$ (left 1), $v_2 = 2$ (up 2). $(-2 \times -1) + (-1 \times 2)$ $= 2 - 2$ $= 0$ So, angle J is a right angle.

We can check angles K and L in the same way. They will also be right angles.

$\boxed{\text{Shape 3 is a square.}}$

Answer

We have drawn two rotated squares and one rotated rectangle on the dot grid.

(i) Square 1 (ABCD) has all four sides equal in length and all four angles are right angles. (ii) Rectangle 1 (EFGH) has opposite sides equal in length and all four angles are right angles. (iii) Square 2 (IJKL) has all four sides equal in length and all four angles are right angles.