one side of a parallelogram has endpoints (2,1) and (3,5) The endpoints for the opposite side are (5,2) and (6,6)
true or false
(2,1)+(1,4) = (3,5)
(5,2)+(1,4) = (6,6)
So, true
True.
To determine whether the statement is true or false, we need to check if the opposite sides of the parallelogram have the same length and are parallel to each other.
To find the length of a side, we can use the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we can find the lengths of both sides.
For the first side with endpoints (2, 1) and (3, 5),
d1 = sqrt((3 - 2)^2 + (5 - 1)^2)
= sqrt(1 + 16)
= sqrt(17)
For the second side with endpoints (5, 2) and (6, 6),
d2 = sqrt((6 - 5)^2 + (6 - 2)^2)
= sqrt(1 + 16)
= sqrt(17)
Now, comparing the lengths of the two sides, we have d1 = sqrt(17) and d2 = sqrt(17). Since both sides have the same length, we can say they are congruent.
Next, we need to check if their slopes are equal to determine if they are parallel. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
For the first side,
m1 = (5 - 1) / (3 - 2)
= 4 / 1
= 4
For the second side,
m2 = (6 - 2) / (6 - 5)
= 4 / 1
= 4
Both sides have the same slope, which means they are parallel.
Since the opposite sides are congruent and parallel, the statement is true.