Verify that the points are the vertices of a parallelogram.

A ( 2,2,2 ) , B ( 3,4,5 ) , C ( 7,6,3 ) , D ( 8,8,6 )

\overrightarrow { A B } = ( 1,2,3 )

is parallel to

\overrightarrow { C D } = ( 1,2,3 )

\overrightarrow { B D } = ( 1,4,5 )

is parallel to

\overrightarrow { A C } = ( 1,4,5 )

Opposites are parallel and same length. So

A B C D

form a parallelogram.
B)

\overrightarrow { A B } = ( 1,2,3 )

is parallel to

\overrightarrow { C D } = ( 5,4,1 )

\overrightarrow { B D } = ( 5,4,1 )

is parallel to

\overrightarrow { A C } = ( 1,2,3 )

Opposites are parallel and same length. So

A B C D

form a parallelogram.
C)

\overrightarrow { A B } = ( 1,2,3 )

is not parallel to

\overrightarrow { C D } = ( 1,2,3 )

\overrightarrow { B D } = ( 5,4,1 )

is not parallel to

\overrightarrow { A C } = ( 5,4,1 )

Opposites are not parallel and same length. So

A B C D

form a parallelogram.
D)

\overrightarrow { A B } = ( 1,2,3 )

is perpendicular to

\overrightarrow { C D } = ( 1,2,3 )

\overrightarrow { B D } = ( 5,4,1 )

is perpendicular to

\overrightarrow { A C } = ( 5,4,1 )

Opposites are perpendicular and same length. So

A B C D

form a parallelogram .
E)

\overrightarrow { A B } = ( 1,2,3 )

is parallel to

\overrightarrow { C D } = ( 1,2,3 )

\overrightarrow { B D } = ( 5,4,1 )

is parallel to

\overrightarrow { A C } = ( 5,4,1 )

Opposites are parallel and same length. So

A B C D

form a parallelogram.

Find the Area of the Triangle with the Given Vertices A(8,−3,−3),B(8,−8,−9),C(−2,6,−6)A ( 8 , - 3 , - 3 ) , B ( 8 , - 8 , - 9 ) , C ( - 2,6 , - 6 )A(8,−3,−3),B(8,−8,−9),C(−2,6,−6)

Multiple Choice