What is the equation for the line of reflection that maps the trapezoid onto itself?

Solution:

We have to find the equation for the line of reflection that maps the trapezoid onto itself.

It is apparent that we can have a line of reflection that maps a trapezoid onto itself,

1. only when the trapezoid has the two non-parallel sides are equal in length and angles they make with any of the parallel sides too are equal (in fact the first condition in a trapezium leads to second) i.e. an isosceles trapezoid. Here, it is so and hence we have a line of reflection that maps a trapezoid onto itself.
2. Further such a line of reflection would be perpendicular to the parallel sides and as here parallel sides are parallel to x-axis, the line of reflection would be parallel to y-axis i.e. of the form x = a
3. Further as every point and its reflection is equidistant from the line of reflection, the line of reflection must pass through the midpoints of parallel sides.

The line that maps a figure onto itself is a line of symmetry of the figure.

From the given figure, the midpoints of parallel sides are (1, 4) and (-2, -2)

The line of symmetry of the trapezoid is x = -2

Therefore, the equation for the line of reflection is x = -2.

What is the equation for the line of reflection that maps the trapezoid onto itself?

Summary:

The equation for the line of reflection that maps the trapezoid onto itself is x = -2.