Math / Geometry

Question6. REASONING A trapezoid has vertices $A(-6,-2), B(-3,-2), C(-1,-4)$ , and $D(-6,-4)$ . a. Rotate the trapezoid $180^{\circ}$ about the origin. What are the coordinates of the image? b. Describe a way to obtain the same image without using rotations.

Studdy Solution

STEP 1

1. The trapezoid is defined by the vertices $A(-6,-2), B(-3,-2), C(-1,-4),$ and $D(-6,-4)$ .
2. The rotation is about the origin.
3. A $180^\circ$ rotation about the origin changes the coordinates of a point $(x, y)$ to $(-x, -y)$ .

STEP 2

1. Perform a $180^\circ$ rotation for each vertex.
2. Describe an alternative transformation to achieve the same result.

STEP 3

Perform a $180^\circ$ rotation for each vertex:
- For $A(-6, -2)$ , the new coordinates are $(6, 2)$ . - For $B(-3, -2)$ , the new coordinates are $(3, 2)$ . - For $C(-1, -4)$ , the new coordinates are $(1, 4)$ . - For $D(-6, -4)$ , the new coordinates are $(6, 4)$ .

STEP 4

Describe an alternative transformation:
To obtain the same image without using rotations, reflect the trapezoid across both the x-axis and the y-axis. This is equivalent to changing the sign of both the x and y coordinates for each vertex.
The coordinates of the image after a $180^\circ$ rotation are: - $A'(6, 2)$ - $B'(3, 2)$ - $C'(1, 4)$ - $D'(6, 4)$
An alternative transformation is a reflection across both the x-axis and the y-axis.

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