Question

a. explain copy this rectangle on your paper, and show that it has only 2 lines of symmetry. why arent the diagonals of a rectangle lines of symmetry?
b. the y - axis is a line of symmetry for a triangle. the coordinates of two of its vertices are (0, 1) and (3, 4). what are the coordinates of the third vertex?
c. which of these letters have rotational symmetry? which have reflective symmetry?
v w x y
d. which of these polygons does not necessarily have rotational symmetry? sketch an example of the polygon without rotational symmetry.
rectangle square
triangle parallelogram

Explanation:

Part a

Step1: Recall Line of Symmetry

A line of symmetry divides a figure into two congruent halves that mirror each other. For a rectangle, let's consider its sides.

Step2: Identify Symmetry Lines

• Horizontal Line: A horizontal line through the midpoints of the vertical sides. Folding along this line maps the top half to the bottom half, and they are congruent.
• Vertical Line: A vertical line through the midpoints of the horizontal sides. Folding along this line maps the left half to the right half, and they are congruent.

Step3: Analyze Diagonals

For a diagonal of a rectangle, folding along a diagonal does not map the rectangle onto itself. The two triangles formed by a diagonal are congruent (by SSS, since opposite sides of a rectangle are equal and the diagonal is common), but the orientation is different. When folded along a diagonal, the vertices do not coincide in a way that the figure is symmetric (e.g., the angles and sides do not align to form the original rectangle). So diagonals are not lines of symmetry.

Step1: Recall Symmetry over y - axis

If a figure is symmetric about the \(y\) - axis, then for a point \((x,y)\) on the figure, the point \((-x,y)\) is also on the figure.

Step2: Analyze Given Points

We have two vertices: \((0,1)\) (which lies on the \(y\) - axis, so its mirror image is itself) and \((3,4)\). The mirror image of \((3,4)\) about the \(y\) - axis is \((- 3,4)\).

Step3: Determine the Third Vertex

Since it's a triangle and symmetric about the \(y\) - axis, the three vertices must be \((0,1)\), \((3,4)\), and \((-3,4)\).

Step1: Recall Rotational Symmetry

A figure has rotational symmetry if it can be rotated by an angle (less than \(360^{\circ}\)) about a center and map onto itself.

• For letter \(V\): Rotating \(V\) by any angle (less than \(360^{\circ}\)) other than \(360^{\circ}\) does not map it onto itself. So no rotational symmetry. It has reflective symmetry (a vertical line through its vertex and base mid - point).
• For letter \(W\): Rotating \(W\) by any angle (less than \(360^{\circ}\)) other than \(360^{\circ}\) does not map it onto itself. So no rotational symmetry. It has reflective symmetry (a vertical line through its center).
• For letter \(X\): Rotating \(X\) by \(180^{\circ}\) about its center maps it onto itself. So it has rotational symmetry (order 2). It also has reflective symmetry (two diagonal lines and a vertical/horizontal line? Wait, actually, \(X\) has four lines of reflective symmetry? No, for the purpose of basic symmetry: vertical, horizontal, and two diagonals? But at least it has reflective symmetry.
• For letter \(Y\): Rotating \(Y\) by any angle (less than \(360^{\circ}\)) other than \(360^{\circ}\) does not map it onto itself. So no rotational symmetry. It has reflective symmetry (a vertical line through its vertex and base mid - point).

Step2: Recall Reflective Symmetry

A figure has reflective symmetry if it can be reflected over a line and map onto itself.

• \(V\): Reflect over vertical line through its axis. Symmetric.
• \(W\): Reflect over vertical line through its center. Symmetric.
• \(X\): Reflect over vertical, horizontal, or diagonal lines. Symmetric.
• \(Y\): Reflect over vertical line through its axis. Symmetric.

Answer:

A rectangle has 2 lines of symmetry (horizontal through mid - vertical sides, vertical through mid - horizontal sides). Diagonals aren't lines of symmetry because folding along a diagonal doesn't map the rectangle onto itself (vertices/parts don't align to recreate the rectangle).

Part b