One side of a parallelogram has endpoints (3, 3) and (1, 7).
What are the endpoints for the side opposite?
(8, 1) and (6, 7)
(6, 1) and (2, 3)
(6, 1) and (8, 5)
(8, 1) and (6, 5)
(6, 1) and (8, 5)
CLUE: The opposite sides are equal and
parallel. Parallel lines have equal slopes.
(3,3),(1,7). m1 = (7-3)/(1-3) = -2.
(8,1),(6,5). m2 = (5-1)/(6-8) = -2.
To find the endpoints for the side opposite of (3, 3) and (1, 7), we can use the fact that the opposite sides of a parallelogram are parallel and equal in length.
Let's first calculate the length of the given side:
Length = sqrt((1 - 3)^2 + (7 - 3)^2)
= sqrt((-2)^2 + (4)^2)
= sqrt(4 + 16)
= sqrt(20)
= 2 * sqrt(5)
Now, we need to find another side that is parallel to this side and has the same length.
Among the answer choices, only (8, 1) and (6, 5) has the same length as 2 * sqrt(5). Therefore, the endpoints for the side opposite of (3, 3) and (1, 7) are (8, 1) and (6, 5).
To find the endpoints of the side opposite to a given side of a parallelogram, we can use the property that opposite sides of a parallelogram are congruent.
Given the endpoints of one side of the parallelogram as (3, 3) and (1, 7), we can calculate the length and slope of this side.
Length of the side = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(1 - 3)^2 + (7 - 3)^2]
= √[(-2)^2 + 4^2]
= √[4 + 16]
= √20
= 2√5
Slope of the side = (y2 - y1) / (x2 - x1)
= (7 - 3) / (1 - 3)
= 4 / (-2)
= -2
Now, we can use the length and slope of the given side to determine the endpoints of the side opposite to it.
1) (8, 1) and (6, 7):
Calculate the length and slope of this side:
Length = √[(6 - 8)^2 + (7 - 1)^2]
= √[(-2)^2 + 6^2]
= √[4 + 36]
= √40
= 2√10
Slope = (7 - 1) / (6 - 8)
= 6 / (-2)
= -3
Since the lengths (2√5 and 2√10) and slopes (-2 and -3) are not equal, this option is incorrect.
2) (6, 1) and (2, 3):
Calculate the length and slope of this side:
Length = √[(2 - 6)^2 + (3 - 1)^2]
= √[(-4)^2 + 2^2]
= √[16 + 4]
= √20
= 2√5
Slope = (3 - 1) / (2 - 6)
= 2 / (-4)
= -1/2
Since the lengths (2√5 and 2√5) and slopes (-2 and -1/2) are not equal, this option is incorrect.
3) (6, 1) and (8, 5):
Calculate the length and slope of this side:
Length = √[(8 - 6)^2 + (5 - 1)^2]
= √[2^2 + 4^2]
= √[4 + 16]
= √20
= 2√5
Slope = (5 - 1) / (8 - 6)
= 4 / 2
= 2
Since the lengths (2√5 and 2√5) and slopes (-2 and 2) are not equal, this option is incorrect.
4) (8, 1) and (6, 5):
Calculate the length and slope of this side:
Length = √[(6 - 8)^2 + (5 - 1)^2]
= √[(-2)^2 + 4^2]
= √[4 + 16]
= √20
= 2√5
Slope = (5 - 1) / (6 - 8)
= 4 / (-2)
= -2
Since the lengths (2√5 and 2√5) and slopes (-2 and -2) are equal, this option is correct.
Therefore, the endpoints for the side opposite to the given side are (8, 1) and (6, 5). So the correct answer is option (4) (8, 1) and (6, 5).