quadrilateral A B C D on a graph. Point A is at ( -5, -1), Point B is at ( 6, 1), Point C is at (4, -3), Point D is at (-7, -5).
Prove the quadrilateral is a parallelogram by using Theorem 5-7; if the diagonals of a quadrilateral bisect each other, then it is a parallelogram. (Hint: Use the Midpoint formula.)
well, what are the diagonals?
AC and BD
midpoint of AC = (-1/2,-2)
midpoint of BD = (-1/2,-2)
Looks like the diagonals bisect each other.
To prove that quadrilateral ABCD is a parallelogram using Theorem 5-7, which states that if the diagonals of a quadrilateral bisect each other, then it is a parallelogram, we need to show that the diagonals of the quadrilateral bisect each other.
Step 1: Find the midpoint of the diagonals.
To find the midpoint of a line segment given two points, you can use the midpoint formula:
Midpoint(x, y) = ((x1 + x2) / 2, (y1 + y2) / 2).
Let's find the midpoints of the diagonals AC and BD separately.
Midpoint of AC:
Point A = (-5, -1)
Point C = (4, -3)
Midpoint(x, y) = ((-5 + 4) / 2, (-1 + -3) / 2)
= (-1/2, -2)
Midpoint of BD:
Point B = (6, 1)
Point D = (-7, -5)
Midpoint(x, y) = ((6 + (-7)) / 2, (1 + (-5)) / 2)
= (-1/2, -2)
Step 2: Compare the midpoints.
Since the midpoint of AC (-1/2, -2) is the same as the midpoint of BD (-1/2, -2), we can conclude that the diagonals of the quadrilateral bisect each other.
Step 3: State the conclusion.
According to Theorem 5-7, if the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Since we have shown that the diagonals of quadrilateral ABCD bisect each other, we can conclude that ABCD is a parallelogram.