E11E11 make generalizations make generalizations about the rotational symmetry about the rotational symmetry property of squares and property of squares and rectangles and apply them.rectangles and apply them.

Let’s Review: What are the Let’s Review: What are the properties of squares that you properties of squares that you already know?already know?

A square is a quadrilateral.A square is a quadrilateral. A square has four congruent side lengths.A square has four congruent side lengths. A square has four equal angles.A square has four equal angles. A square is a special type of rhombus.A square is a special type of rhombus. A square is a parallelogram.A square is a parallelogram. Each angle in a square measures 90 degrees.Each angle in a square measures 90 degrees. The diagonals of a square are equal in length; bisect The diagonals of a square are equal in length; bisect

each other; intersect to form four right angles and each other; intersect to form four right angles and combined with the previous properties this means combined with the previous properties this means they are perpendicular-bisectors of each other; are they are perpendicular-bisectors of each other; are bisectors of the vertex angles of the square, thus bisectors of the vertex angles of the square, thus forming 45 degree angles; and form four congruent forming 45 degree angles; and form four congruent isosceles right trianglesisosceles right triangles

Now let’s investigate another Now let’s investigate another property of squares:property of squares:

Use a square from the pattern blocks, and mark Use a square from the pattern blocks, and mark one of its vertices with a chalk dot.one of its vertices with a chalk dot.

Next, carefully trace the block to make a square on Next, carefully trace the block to make a square on a sheet of paper. a sheet of paper.

With the square block placed inside its picture, With the square block placed inside its picture, rotate it clockwise with the centre of rotation being rotate it clockwise with the centre of rotation being the centre of the square (intersection point of its the centre of the square (intersection point of its two diagonals) until it perfectly matches its picture two diagonals) until it perfectly matches its picture again. again.

Notice that the marked vertex is at the next corner. Notice that the marked vertex is at the next corner. Repeat this rotation. How many times does the Repeat this rotation. How many times does the

square appear in four identical positions during one square appear in four identical positions during one complete 360-degree rotation? complete 360-degree rotation?

The answer is 4, and the square is said to have The answer is 4, and the square is said to have rotational symmetry of order 4.rotational symmetry of order 4.

Rotational Symmetry of a Rotational Symmetry of a SquareSquare

Let’s Review: What are the Let’s Review: What are the properties of rectangles that properties of rectangles that you already know?you already know?

A rectangle is a quadrilateral.A rectangle is a quadrilateral. A rectangle has four equal angles.A rectangle has four equal angles. A rectangle is a special type of parallelogram with A rectangle is a special type of parallelogram with

all 90- degree angles.all 90- degree angles. The diagonals of a rectangle are are equal in The diagonals of a rectangle are are equal in

length; bisect each other; form two pairs of equal length; bisect each other; form two pairs of equal opposite angles at the point of intersection; form opposite angles at the point of intersection; form two angles at each vertex of the rectangle that sum two angles at each vertex of the rectangle that sum to 90 degrees and have the same measures as the to 90 degrees and have the same measures as the two angles at the other vertices; andtwo angles at the other vertices; andform two pairs of congruent isosceles trianglesform two pairs of congruent isosceles triangles

Now let’s investigate another Now let’s investigate another property of rectangles:property of rectangles:

Make a rectangle from hard paper, and mark one of Make a rectangle from hard paper, and mark one of its vertices with a chalk dot.its vertices with a chalk dot.

Next, carefully trace the rectangle to make a Next, carefully trace the rectangle to make a second rectangle on a sheet of paper. second rectangle on a sheet of paper.

With the rectangle block placed inside its picture, With the rectangle block placed inside its picture, rotate it clockwise with the centre of rotation being rotate it clockwise with the centre of rotation being the centre of the rectangle (intersection point of its the centre of the rectangle (intersection point of its two diagonals) until it perfectly matches its picture two diagonals) until it perfectly matches its picture again. again.

Notice that the marked vertex is at the next corner. Notice that the marked vertex is at the next corner. Repeat this rotation. How many times does the Repeat this rotation. How many times does the

rectangle appear in four identical positions during rectangle appear in four identical positions during one complete 360-degree rotation? one complete 360-degree rotation?

The answer is 2, and the rectangle is said to have The answer is 2, and the rectangle is said to have rotational symmetry of order 4.rotational symmetry of order 4.

Rotational Symmetry of a Rotational Symmetry of a RectangleRectangle

http://regentsprep.org/Regents/math/quad/LQuad.htm

Meet Some of the Members of the Quadrilateral Family

Student Activities:Student Activities:

E11.1E11.1 Investigate to find out if a Investigate to find out if a square is the only quadrilateral with square is the only quadrilateral with rotational symmetry of order 4.rotational symmetry of order 4.

E11.2E11.2 Investigate what other Investigate what other quadrilaterals besides rectangles have quadrilaterals besides rectangles have rotational symmetry of order 2.rotational symmetry of order 2.

Which ones also have two lines of Which ones also have two lines of reflective symmetry?reflective symmetry?