Question

2 here is segment de. translate segment de by directed line segment w. label the new endpoints d and e. a. connect d to d and e to e. b. what kind of shape did you draw? what properties does it have? explain your reasoning.

Explanation:

Step1: Recall translation property

Translation is a rigid - motion that moves every point of a figure the same distance in the same direction. When we translate segment $DE$ by directed line segment $w$ to get $D'E'$, then $DD'\parallel EE'$ and $DD' = EE'$.

Step2: Analyze the shape formed

When we connect $D$ to $D'$ and $E$ to $E'$, along with $DE$ and $D'E'$, we have a quadrilateral. Since $DE\parallel D'E'$ (because translation preserves parallelism) and $DD'\parallel EE'$, the shape drawn is a parallelogram.

Step3: List properties of the parallelogram

In a parallelogram, opposite sides are parallel and equal. So, $DE = D'E'$, $DD' = EE'$, $DE\parallel D'E'$, and $DD'\parallel EE'$. Also, opposite angles are equal, and consecutive angles are supplementary.

Answer:

a. (This part is a construction step which can be done with a straight - edge. Draw lines connecting $D$ to $D'$ and $E$ to $E'$).
b. The shape drawn is a parallelogram.
Properties: Opposite sides are parallel ($DE\parallel D'E'$ and $DD'\parallel EE'$) and equal ($DE = D'E'$, $DD' = EE'$), opposite angles are equal, and consecutive angles are supplementary. Reasoning: Translation is a rigid motion that preserves distance and parallelism. When we translate segment $DE$ to $D'E'$ and connect the corresponding endpoints, we get a quadrilateral with two pairs of parallel sides, which is the definition of a parallelogram.