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?

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:

• There is a vertical line of symmetry that passes through the middle of the horizontal bars and the center of the vertical bars. When we fold the figure along this vertical line, the left and right halves coincide.
• There is also a horizontal line of symmetry that passes through the middle of the vertical bars and the center of the horizontal bars. When we fold the figure along this horizontal line, the upper and lower halves coincide.

Step2: Draw the Lines

We draw a vertical line through the center of the 'H' (along the vertical axis of symmetry) and a horizontal line through the center of the 'H' (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 intersection point of the two lines of symmetry (the vertical and horizontal lines of symmetry). This point is the mid - point of the horizontal bars and also the mid - point of the vertical bars. If we consider the coordinates (assuming a coordinate system), if the top - most point of the upper horizontal bar is \( (x_1,y_1) \), the bottom - most point of the lower horizontal bar is \( (x_1,y_2) \), the left - most point of the left vertical bar is \( (x_3,y_1) \) and the right - most point of the right vertical bar is \( (x_4,y_1) \), the center is at \( (\frac{x_3 + x_4}{2},\frac{y_1 + y_2}{2}) \) (or visually, the middle of the 'H').

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 and still look the same (coincide with the original figure) in a full rotation ( \( 360^{\circ} \)).
• The magnitude of rotational symmetry is the smallest angle through which the figure can be rotated to coincide with itself.

Step2: Analyze the 'H' - shaped Figure

• For the order of rotational symmetry: When we rotate the 'H' - shaped figure by \( 180^{\circ} \) about its center of symmetry, it coincides with itself. If we rotate it by \( 360^{\circ} \), it also coincides with itself. But the smallest non - zero angle of rotation for which it coincides with itself is \( 180^{\circ} \). The number of times it coincides with itself in a \( 360^{\circ} \) rotation: \( \frac{360^{\circ}}{180^{\circ}}=2 \). So the order of rotational symmetry is 2.
• For the magnitude of rotational symmetry: The magnitude is the smallest angle of rotation that maps the figure onto itself. As we found, when we rotate the figure by \( 180^{\circ} \), it coincides with itself, and this is the smallest non - zero angle. So the magnitude is \( 180^{\circ} \).

Answer:

The figure has two lines of symmetry: one vertical line passing through the center (dividing the left and right parts) and one horizontal line passing through the center (dividing the upper and lower parts). (To draw them, sketch a vertical line through the middle of the horizontal segments and a horizontal line through the middle of the vertical segments of the 'H' - shaped figure.)

Question 16