Question

prove that the quadrilateral whose vertices are the points A(-1,1), B(-3,4), C(1,5) and D(3,2) is a parallelogram.

Answer 1

Since
`\overrightarrow{AB} = \overrightarrow{DC}` and
`\overrightarrow{AD} = \overrightarrow{BC}`, then the quadrilateral ABCD is a parallelogram.

Explanation:

First, we label each point of the quadrilateral with the help of a graphing tool. If the quadrilateral ABCD is a parallelogram, then
`\overrightarrow{AB} = \overrightarrow{DC}` and
`\overrightarrow{AD} = \overrightarrow{BC}`. If we know that
`A(x,y) =(-1, 1)`,
`B(x,y) =(-3,4)`,
`C(x,y) = (1,5)` and
`D(x,y) = (3,2)`, then the measure of each vector is, respectively:

`\overrightarrow{AB} = (-3,4)-(-1,1)`

`\overrightarrow{AB} = (-2, 3)`

`\overrightarrow{DC} = (1,5)-(3,2)`

`\overrightarrow{DC} = ( -2,3)`

`\overrightarrow{AD} = (3,2)-(-1,1)`

`\overrightarrow{AD} = (4, 1)`

`\overrightarrow{BC} = (1,5)-(-3,4)`

`\overrightarrow{BC} = (4, 1)`

Since
`\overrightarrow{AB} = \overrightarrow{DC}` and
`\overrightarrow{AD} = \overrightarrow{BC}`, then the quadrilateral ABCD is a parallelogram.