Question

Trapezoid ABCD is shown below.

In a graph, trapezoid ABCD is drawn at points A(-6, -2), B(-6, -6), C(-3, -5), D(-3, -3).

If trapezoid ABCD were reflected across the y-axis, and then translated up 7 units to create trapezoid A'B'C'D', which set of sides would be parallel in the resulting image?

I.
and
II.
and
III.
and
IV.
and
A.
I and IV only
B.
II and III only
C.
IV only
D.
I only

Answer 1

To analyze the trapezoid ABCD reflected across the y-axis and translated up 7 units, we start by determining the coordinates of the vertices after these transformations.

1. **Reflection across the y-axis:**
- Point A(-6, -2) becomes A' (6, -2)
- Point B(-6, -6) becomes B' (6, -6)
- Point C(-3, -5) becomes C' (3, -5)
- Point D(-3, -3) becomes D' (3, -3)

2. **Translation up 7 units:**
- A' (6, -2) becomes A'' (6, -2 + 7) → A'' (6, 5)
- B' (6, -6) becomes B'' (6, -6 + 7) → B'' (6, 1)
- C' (3, -5) becomes C'' (3, -5 + 7) → C'' (3, 2)
- D' (3, -3) becomes D'' (3, -3 + 7) → D'' (3, 4)

The new coordinates of the vertices of trapezoid A''B''C''D'' are:
- A'' (6, 5)
- B'' (6, 1)
- C'' (3, 2)
- D'' (3, 4)

Now we check the slopes of the sides of trapezoid A''B''C''D'' to determine which pairs of sides are parallel.

- **Vertical sides:**
- AB (from A'' to B''): Both points have the same x-coordinate, 6 (vertical line).
- CD (from C'' to D''): Both points have the same x-coordinate, 3 (vertical line).

Since AB and CD are both vertical and have different x-coordinates (6 and 3), they are parallel.

Next, we check the other pairs:
- **Sides A''C'' and B''D'':**
- Slope of A''C'' = (2 - 5) / (3 - 6) = -3 / -1 = 3
- Slope of B''D'' = (4 - 1) / (3 - 6) = 3 / -3 = -1

- **Sides A''D'' and B''C'':**
- Slope of A''D'' = (4 - 5) / (3 - 6) = -1 / -3 = 1/3
- Slope of B''C'' = (2 - 1) / (3 - 6) = 1 / -3

Due to the individual calculations, we conclude that only AB and CD are parallel.

Thus, the answer is **IV only**, which states that AB and CD are the parallel sides.

So the correct answer is **C. IV only**.