Question

refer to the figure at the right.

1. draw the line(s) of symmetry in the figure.
2. locate the center of symmetry for the figure.
3. what is the order and magnitude of symmetry for the figure?

order : 2, magnitude : 180 degrees

Explanation:

Question 15

Step1: Recall Line of Symmetry

A line of symmetry divides a figure into two mirror - image halves. For the letter 'H' - shaped figure, we can find two lines of symmetry. One is a vertical line passing through the middle of the horizontal bar and the middle of the vertical bars. The other is a horizontal line passing through the middle of the vertical bars and the middle of the horizontal bar.

Step2: Draw the Lines

Draw a vertical line through the center of the figure (along the vertical axis of symmetry) and a horizontal line through the center of the figure (along the horizontal axis of symmetry).

Step1: Recall Center of Symmetry

The center of symmetry of a figure is a point such that for every point \(P\) in the figure, there is a corresponding point \(P'\) such that the center is the mid - point of the segment \(PP'\).

Step2: Locate the Center

For the 'H' - shaped figure, the center of symmetry is the point of intersection of the two lines of symmetry (the vertical and horizontal lines of symmetry). It is the mid - point of the horizontal bar and also the mid - point between the two vertical bars.

Step1: Recall Order and Magnitude of Symmetry

The order of rotational symmetry of a figure is the number of times the figure can be rotated about its center of symmetry so that it maps onto itself in a full \(360^{\circ}\) rotation. The magnitude of rotational symmetry is the smallest angle through which the figure can be rotated to map onto itself.

Step2: Determine Order

When we rotate the 'H' - shaped figure about its center of symmetry, it maps onto itself when rotated by \(180^{\circ}\) and also when rotated by \(360^{\circ}\) (which is the same as the original position). But the non - trivial rotation (other than \(360^{\circ}\)) that maps it onto itself is \(180^{\circ}\). The number of times it maps onto itself in a \(360^{\circ}\) rotation: \(\frac{360^{\circ}}{180^{\circ}}=2\). So the order of rotational symmetry is 2.

Step3: Determine Magnitude

The smallest angle of rotation that maps the figure onto itself is \(180^{\circ}\), so the magnitude of rotational symmetry is \(180^{\circ}\).

Answer:

There are two lines of symmetry: one vertical line through the center and one horizontal line through the center. (To be drawn on the figure: a vertical line and a horizontal line intersecting at the center of the 'H' - shaped figure)

Question 16