Which transformation carries the trapezoid onto itself? A. a reflection across the line y = 0
B. a rotation of 180° about the point (- 1, 0)
C. a rotation of 180 ° about the origin
D. a reflection across the line x = - 1 -1

D. a reflection across the line x = - 1

Step-by-step explanation:

First, you can see that the lengths of the top and the base of the figure are different, so we can discard the rotations of 180°, because those will give a different figure.

Now, for reflections, we need to reflect the image around a line of symmetry.

A line of symmetry means that the image at the right of the figure is an exact reflex of the image at the left of the figure, for this particular case, we can find a line x = c. Such that the distance between c and the x-component of the top vertexes is the same for both of them.

(Again, because the top and the bottom have different lengths, there is no line y = b such that a reflection across that line will leave the figure invariant)

The left top vertex is at x = -3

The right top vertex is at x = 1.

We can find the mid-value between them as:

c = (-3 - 1)/2 = -2/2 = -1

Then:

-3 - c = -3 - (-1) = -3 + 1 = -2

1 - c = 1 - (-1) = 2

While the numbers are different (in sign) as those represent distance between points, we only care for the absolute value, in that case:

I-2I = I2I

Then the distances are exactly the same, and we can take c = -1

This means that a reflection about the line x = -1 carries the figure onto itself.

Then the correct option is:

D. a reflection across the line x = - 1