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Chapter 16: Problem 1488

If three vertices of rhombus are \((6,0,1)(8,-3,7)\) \((2,-5,10)\), then forth vertices \(=\) (A) \((0,-2,-4)\) (B) \((0,-2,4)\) (C) \((0,2,4)\) (D) \((0,2,-4)\)

Short Answer

Expert verified
The fourth vertex of the rhombus is (B) \((0,-2,4)\).

Step by step solution

01

Find the Midpoint of Vertex A and C

To find the fourth vertex, we will use the fact that the diagonals of a rhombus bisect each other at a right angle. So the midpoint of A and C will be the same as the midpoint of B and D. Let's find the midpoint of vertex A and C using the midpoint formula: M = \(\frac{A+C}{2}\) M = \(\frac{(6,0,1)+(2,-5,10)}{2}\) M = \(\frac{(8,-5,11)}{2}\) M = (4,-2.5,5.5)
02

Express Fourth Vertex D as Variable

Let's denote D(x, y, z) as the fourth vertex, not knowing its coordinate values yet. This will help us to create an equation system to solve to find the actual coordinate values.
03

Use the Midpoint Formula for Vertex B and D

Now, we will find the midpoint of vertex B and D using the same midpoint formula as before. We will equate the midpoint of B and D with the midpoint of A and C: Midpoint of B and D = \(\frac{(8,-3,7)+(x,y,z)}{2} = (4,-2.5,5.5)\) Equating the coordinates, we get the following equations: \(\frac{8+x}{2} = 4\), \(\frac{-3+y}{2} = -2.5\), \(\frac{7+z}{2} = 5.5\)
04

Solve the Equations for x, y, and z

Solve the three equations to find the coordinates of the fourth vertex D. \(8+x = 8\): x = 0 \(-3+y = -5\): y = -2 \(7+z = 11\): z = 4 Thus, the fourth vertex D is (0, -2, 4). The correct answer is (B) \((0,-2,4)\).

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