Question

Consider the effect of the transformation (x, y) → (x, 2y) on the parallelogram ABCD with vertices A(0, 0), B(1, 1), C(3, 1), and D(2, 0). Select True or False for each statement.

Answer 1

The transformation (x, y) → (x, 2y) doubles the y-coordinates of the parallelogram ABCD vertices, resulting in a new parallelogram with doubled height and consequently doubled area, demonstrating that the change in height is proportional to the original height.

Step-by-step explanation:

The question posed relates to the effect of a transformation on the coordinates of a parallelogram's vertices. Specifically, we look at the transformation (x, y) → (x, 2y), which stretches the y-coordinates of the vertices while keeping the x-coordinates unchanged. For the given parallelogram ABCD with vertices A(0, 0), B(1, 1), C(3, 1), and D(2, 0), we apply the transformation to each vertex:

• Vertex A(0, 0) becomes A'(0, 0 × 2) = A'(0, 0)
• Vertex B(1, 1) becomes B'(1, 1 × 2) = B'(1, 2)
• Vertex C(3, 1) becomes C'(3, 1 × 2) = C'(3, 2)
• Vertex D(2, 0) becomes D'(2, 0 × 2) = D'(2, 0)

The transformation doubles the height (y-coordinate) of the parallelogram, while the base (x-coordinate) remains the same. Consequently, the area of the parallelogram also doubles, confirming that the change in height is proportional to the original height. By applying the transformation to a simple geometric figure, students can visualize and understand the properties of transformations and their effects on the shapes.