Sammy is classifying quadrilateral EFGH. How can she use coordinate geometry to determine whether EFGH is a parallelogram?

Question

Sammy is classifying quadrilateral EFGH. How can she use coordinate geometry to determine whether EFGH is a parallelogram?

E(0, 4), F(0, 7), G(1, 5), H(1, 2) 

Sammy can use the distance formula to show that the opposite sides have equal slopes.

Sammy can use the distance formula to show that all sides are equal. 

Sammy can use the slope formula to show that the opposite sides have equal slopes. 

Sammy can use the slope formula to show that all sides are equal.

Answer 1

I am assuming the 4 final statements in your question are the possible answers.

"Sammy can use the distance formula to show that the opposite sides have equal slopes." - The distance formula is not used to find the slopes of lines, therefore this is not a possible answer.

"Sammy can use the distance formula to show that all sides are equal." - The distance formula can be used to find the lengths of the sides, but this is a parallelogram - all sides are not necessarily equal, so this won't help to determine if it is parallelogram

"Sammy can use the slope formula to show that the opposite sides have equal slopes." - This is useful - angles that are diagonal in a parallelogram are congruent, which means that the slopes of the opposite sides are the same slope (opposite sides are parallel) by the converse to the alternate interior angles theorem

"Sammy can use the slope formula to show that all sides are equal." - The slope formula can't be used to find the lengths of the sides, and as previously indicated, the sides are not necessarily equal, so this won't determine if the shape is a parallelogram.