One side of a parallelogram has endpoints #(3,3), (1,7)#. What are the endpoints for the side opposite?

Question

One side of a parallelogram has endpoints #(3,3), (1,7)#.  What are the endpoints for the side opposite?

Noah Atkinson · Accepted Answer

There are an infinite number of parallelograms with opposite side #(3,3),(1,7)#. Any non-collinear segment with endpoints #(3+x,3+y),(1+x,7+y)# will work, which we can rewrite

#(x,y),(x-2,y+4)#

Sadie Allen · Answer

undefined To find the endpoints of the side opposite in the parallelogram, you can use the property that opposite sides of a parallelogram are parallel and equal in length.

Given the endpoints of one side as (3, 3) and (1, 7), the vector representing this side is $\langle 1 - 3, 7 - 3 \rangle = \langle -2, 4 \rangle$.

Since opposite sides of a parallelogram have the same vector, the other side's vector will also be $\langle -2, 4 \rangle$.

To find the endpoints of the side opposite, you start with one of the given endpoints, say (3, 3), and add the components of the vector $\langle -2, 4 \rangle$ to it.

So, adding $-2$ to the x-coordinate and $4$ to the y-coordinate of (3, 3), you get the other endpoint: $(3 - 2, 3 + 4) = (1, 7)$.

Hence, the endpoints of the side opposite to the given side are (1, 7) and (3 - 2, 3 + 4), which is (1, 7).