Use the following information and diagram to answer the question.
Given: Quadrilateral ABDC has vertices at A(2,6), B(6,8), C(1,2), and D(5,4).
Prove: Quadrilateral ABDC is a parallelogram.
Which plan for a proof for this problem will show that quadrilateral ABDC is a parallelogram?
Responses
Why not just show that opposite sides are parallel, that is, they have
the same slope ? That would be so easy to do.
After all, wouldn't it make sense to use the definition of "parallelogram" ?
Plan A: Show that opposite sides of the quadrilateral are congruent.
Plan B: Show that the diagonals of the quadrilateral bisect each other.
Use the following information and diagram to answer the question.
Given: Quadrilateral ABDC has vertices at A(2,6), B(6,8), C(1,2), and D(5,4).
Prove: Quadrilateral ABDC is a parallelogram.
Which plan for a proof for this problem will show that quadrilateral ABDC is a parallelogram?
A. Use the midpoint formula to find the midpoint of AD¯¯¯¯¯¯¯¯ and BC¯¯¯¯¯¯¯¯. Show that the midpoint for both segments is the same. Since the diagonals of a quadrilateral intersect each other, then the quadrilateral is a parallelogram.
B.Use the midpoint formula to find the midpoint of AB¯¯¯¯¯¯¯¯ and CD¯¯¯¯¯¯¯¯. Show that the midpoint of these two sides has the same x-coordinate. Then, show that the midpoint of BD¯¯¯¯¯¯¯¯ and AC¯¯¯¯¯¯¯¯ has the same y-coordinate. Thus, the quadrilateral is a parallelogram.
C.Use the midpoint formula to find the midpoint of AD¯¯¯¯¯¯¯¯ and BC¯¯¯¯¯¯¯¯. Show that the midpoint for both segments is the same. By the definition of midpoint and bisect, AD¯¯¯¯¯¯¯¯ and BC¯¯¯¯¯¯¯¯ bisect each other. Since the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
D.Use the midpoint formula to find the midpoint of AB¯¯¯¯¯¯¯¯ and CD¯¯¯¯¯¯¯¯. Show that the midpoint for both segments is the same. Since opposite sides of a quadrilateral have the same midpoint, then the quadrilateral is a parallelogram.
Yes, so the answerer is
Use the midpoint formula to find the midpoint of AD¯¯¯¯¯¯¯¯ and BC¯¯¯¯¯¯¯¯. Show that the midpoint for both segments is the same. By the definition of midpoint and bisect, AD¯¯¯¯¯¯¯¯ and BC¯¯¯¯¯¯¯¯ bisect each other. Since the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
To prove that quadrilateral ABDC is a parallelogram, one possible plan for a proof is to show that opposite sides are parallel.
To do this, we can use the slope formula to calculate the slopes of the opposite sides of the quadrilateral. If the slopes are equal, then the sides are parallel.
Here's a step-by-step plan to prove the parallelogram:
1. Calculate the slope of side AB using the formula:
slope_AB = (y2 - y1) / (x2 - x1)
slope_AB = (8 - 6) / (6 - 2)
2. Calculate the slope of side CD using the same formula:
slope_CD = (y2 - y1) / (x2 - x1)
slope_CD = (4 - 2) / (5 - 1)
3. Calculate the slope of side BC and the slope of side AD in the same way.
4. If the slopes of opposite sides are equal, then the quadrilateral is a parallelogram. Therefore, compare the slopes:
slope_AB = slope_CD
slope_BC = slope_AD
If all slopes are equal, quadrilateral ABDC is a parallelogram.
You can use these calculations to answer the question and prove the parallelogram.