Verify the following: (–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, … - Vidyadip

Question

Verify the following: (–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.

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Answer

Let (–1, 2, 1), (1, –2, 5), (4, –7, 8), and (2, –3, 4) be denoted by A, B, C, and D respectively. $\text{AB}=\sqrt{(1+1)^2+(-2-2)^2+(5-1)^2}$ $=\sqrt{4+16+16}$ $=\sqrt{36}$ $=6$ $\text{BC}=\sqrt{(4-1)^2+(-7+2)^2+(8-5)^2}$ $=\sqrt{9+25+9}=\sqrt{43}$ $\text{CD}=\sqrt{(2-4)^2+(-3+7)^2+(4-8)^2}$ $=\sqrt{4+16+16}$ $=\sqrt{36}$ $=6$ $\text{DA}=\sqrt{(-1-2)^2+(2+3)^3+(1-4)^2}$ $=\sqrt{9+25+9}=\sqrt{43}$ Here, $\text{AB = CD = 6, BC = AD =}\sqrt{43}$ Hence, the opposite sides of quadrilateral ABCD, whose vertices are taken in order, are equal. Therefore, ABCD is a parallelogram. Hence, the given points are the vertices of a parallelogram.

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