Question

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

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

Step-by-step solution

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)

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.

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: \(\frac{8+x}{2} = 4\), \(\frac{-3+y}{2} = -2.5\), \(\frac{7+z}{2} = 5.5\)

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) \((0,-2,4)\).

Key concepts

Midpoint Formula

The midpoint formula is a crucial tool in geometry that helps us find the exact center point between two known points. Imagine you have two points, say \(A(x_1, y_1, z_1)\) and \(C(x_2, y_2, z_2)\), in a 3D space. The midpoint \(M(x_m, y_m, z_m)\) is obtained by averaging the corresponding coordinates of these points:

• The formula is: \((x_m, y_m, z_m) = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)\).

This formula is essential, particularly for symmetric shapes like rhombuses, where diagonals bisect each other and split into equal halves intersecting at the midpoint.
In the exercise, we use it twice: once for the vertices \(A\) and \(C\), and again for \(B\) and \(D\). This helps confirm the position of the diagonals' intersection point.

Rhombus Properties

Rhombuses are interesting quadrilaterals with some special properties that set them apart. Here are some key characteristics:

• All sides are of equal length, making every rhombus equilateral.
• The diagonals of a rhombus bisect each other at right angles, meaning they cross at 90 degrees and divide each diagonal into two equal parts.
• Diagonals also bisect opposite angles, further highlighting symmetry.

Understanding these properties is essential for solving geometry problems involving rhombuses, like finding a missing vertex.
In our exercise, we utilized the property that the diagonals bisect each other. By finding the same midpoint from different vertices, we ensure the shape's symmetry, crucial for accurately determining the unknown corner.

3D Coordinate Geometry

3D coordinate geometry extends the principles of 2D geometry into the third dimension. It allows us to define points in a three-dimensional space using an ordered triple \(x, y, z\). Here, \(x\), \(y\), and \(z\) represent the point's position along each axis.

• The \(x\) axis usually represents the horizontal direction.
• The \(y\) axis covers vertical movements.
• The \(z\) axis determines depth or the distance into or out of the page.

This framework is vital for accurately mapping and solving problems involving spatial relationships, like with the rhombus in our exercise. It allows us to find midpoints, distances, and angles in space, crucial for determining unknown vertices like in the given problem.
By using this system, we can effectively navigate the complexities of 3D shapes and ensure we're visualizing and calculating positions accurately.