Answer to Question #146836 in Analytic Geometry for Dhruv rawat

Question #146836
Show that the co planar points A(2,3,2), B(4,7,6), C(1,2,3), D(-1,-2,-1) form a parallelogram
1
Expert's answer
2020-11-26T19:40:02-0500

Require to show that the given co-planar points A(2,3,2), B(4,7,6), C(1,2,3), and D(-1,-2,-1)form a parallelogram.

Find the lengths of the sides AB, BC, CD and AD of the quadrilateral ABCDABCD

Now

AB=(42)2+(73)2+(62)2=4+16+16=6AB=\sqrt{(4-2)^2+(7-3)^2+(6-2)^2}=\sqrt{4+16+16}=6

BC=(14)2+(27)2+(36)2=9+25+9=43BC=\sqrt{(1-4)^2+(2-7)^2+(3-6)^2}=\sqrt{9+25+9}=\sqrt{43}

CD=(11)2+(22)2+(13)2=4+16+16=6CD=\sqrt{(-1-1)^2+(-2-2)^2+(-1-3)^2}=\sqrt{4+16+16}=6

AD=(12)2+(23)2+(12)2=9+25+9=43AD=\sqrt{(-1-2)^2+(-2-3)^2+(-1-2)^2}=\sqrt{9+25+9}=\sqrt{43}

Observe that AB=CDAB=CD and BC=ADBC=AD

That is opposite sides of the quadrilateral ABCDABCD are equal.

That is the quadrilateral ABCDABCD is a parallelogram.

Therefore, the given co-planar points form a parallelogram.




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