Question

PLZ HELP -geometry- Find the coordinates of the intersection of the diagonals of parallelogram GHJK with vertices G(1,3), H(4,3), J(5,1), and K(2,1).

The coordinates are ( , ).

Answer 1

The coordinates of the intersection of the diagonal of the parallelogram are
`M(x,y) = (3,2)`.

Explanation:

Diagonals are represented by line segments GJ and KH. Since quadrilateral is a parallelogram, then coordinates of the intersection are located at midpoint of each diagonal (
`M(x,y)`). That is:

`M(x,y) = G(x,y) + (1)/(2)\cdot \overrightarrow {GJ}`

`M(x,y) = G(x,y) +(1)/(2)\cdot [J(x,y)-G(x,y)]` (1)

If we know that
`G(x,y) = (1,3)` and
`J(x,y) = (5,1)`, the coordinates of the intersection of the diagonals of the parallelogram are:

`M(x,y) = (1,3) +(1)/(2)\cdot [(5,1)-(1,3)]`

`M(x,y) = (1,3) +(1)/(2)\cdot (4,-2)`

`M(x,y) = (1,3) +(2,-1)`

`M(x,y) = (3,2)`

There is another form:

`M(x,y) = K(x,y) +(1)/(2)\cdot \overrightarrow{KH}`

`M(x,y) = K(x,y) + (1)/(2)\cdot [H(x,y)-K(x,y)]` (2)

If we know that
`K(x,y) = (2, 1)` and
`H(x,y) = (4,3)`, the coordinates of the intersection of the diagonals of the parallelogram are:

`M(x,y) = (2,1) + (1)/(2)\cdot [(4,3)-(2,1)]`

`M(x,y) = (2,1) + (1)/(2) \cdot (2,2)`

`M(x,y) = (2,1) + (1,1)`

`M(x,y) = (3,2)`

Therefore, the coordinates of the intersection of the diagonal of the parallelogram are
`M(x,y) = (3,2)`.