Question

The points \(O(0,0,0), P(1,4,6),\) and \(Q(2,4,3)\) lie at three vertices of a parallelogram. Find all possible locations of the fourth vertex.

Short answer

Answer: The possible locations for the fourth vertex are (2,8,12), (4,8,6), and (3,4,0).

Step-by-step solution

Case 1: O and P are midpoints of opposite sides

Since O is the origin (0,0,0) and P has coordinates (1,4,6), the midpoint formula can be used to find the coordinates of the fourth vertex R in this case. Let \((x,y,z)\) be the coordinates of R. \(\frac{0 + x}{2} = 1 \Rightarrow x = 2,\) \(\frac{0 + y}{2} = 4 \Rightarrow y = 8,\) \(\frac{0 + z}{2} = 6 \Rightarrow z = 12.\) So, the coordinates of the fourth vertex R are \((2,8,12)\) in this case.

Case 2: O and Q are midpoints of opposite sides

Here, O is again the origin (0,0,0) and Q has coordinates (2,4,3). We can find the coordinates of the fourth vertex S using the midpoint formula. Let \((x',y',z')\) be the coordinates of S. \(\frac{0 + x'}{2} = 2 \Rightarrow x' = 4,\) \(\frac{0 + y'}{2} = 4 \Rightarrow y' = 8,\) \(\frac{0 + z'}{2} = 3 \Rightarrow z' = 6.\) So, the coordinates of the fourth vertex S are \((4,8,6)\) in this case.

Case 3: P and Q are midpoints of opposite sides

In this scenario, P has coordinates (1,4,6) and Q has coordinates (2,4,3). Let's find the coordinates of the fourth vertex T using the midpoint formula. Let \((x'',y'',z'')\) be the coordinates of T. \(\frac{1 + x''}{2} = 2 \Rightarrow x'' = 3,\) \(\frac{4 + y''}{2} = 4 \Rightarrow y'' = 4,\) \(\frac{6 + z''}{2} = 3 \Rightarrow z'' = 0.\) So, the coordinates of the fourth vertex T are \((3,4,0)\) in this case. To summarize, the possible locations for the fourth vertex of the parallelogram are \((2,8,12)\) , \((4,8,6)\), and \((3,4,0)\)

Key concepts

Midpoint Formula

The midpoint formula is a fundamental concept in geometry, particularly useful in coordinate systems. It helps us find a point that is exactly halfway between two other points. This is especially helpful when dealing with shapes like parallelograms where opposite sides are equal and parallel.

In a three-dimensional space, if you have two points, say \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\), the midpoint \(M\) of these points is calculated as:

• \(x_m = \frac{x_1 + x_2}{2}\)
• \(y_m = \frac{y_1 + y_2}{2}\)
• \(z_m = \frac{z_1 + z_2}{2}\)

This formula is straightforward and very powerful.

In the given exercise, this formula is repeatedly used to locate the possible positions of the fourth vertex of a parallelogram. By considering different combinations of points and using the midpoint formula, we easily derive new points, confirming the geometrical property that a pair of opposite vertices can be the midpoint of each other's sides.

Coordinate Geometry

Coordinate geometry lets us use algebraic methods to understand geometric principles. Through this, we can analyze shapes using a coordinate plane or space, incorporating equations and formulas. This branch of geometry offers a clear method for drawing connections between algebraic expressions and geometric figures.

For the exercise, coordinate geometry plays a crucial role in identifying the position of a missing vertex in a parallelogram. By fixing the vertices in numerical coordinates, simple computations are performed to find crucial points like midpoints and unknown vertices. This way, abstract geometric concepts become tangible, and practical solutions can be achieved.

• Use coordinate systems to position geometric shapes.
• Harness algebra to derive unknown elements within set shapes.
• Implement formulas to ensure calculations align with geometric facts.

This understanding merges geometry and math in a very applicable way, highlighting a harmonious blend of theory and computation.

Three-Dimensional Vectors

In the context of this exercise, vectors illustrate how points and shapes move and align inside three-dimensional space. Instead of relying solely on two dimensions (x and y), a third z-dimension adds depth, opening a whole new dimension to calculations and geometric visualizations.

In mathematics, vectors are expressed as \(\mathbf{v} = (x, y, z)\), indicating both direction and magnitude. This is how geometry steps into 3D, and we can interpret not just position but also "movement" by understanding the difference between two points.

• Vectors help determine the positioning of points in 3D spaces.
• They are used to calculate angles, areas, and perimeters in a three-dimensional context.
• With 3D vectors, we define and understand complex shapes beyond the flat two-dimensional plane.

In our case, the parallelogram vertices calculations effectively demonstrate how vectors make it possible to identify third-dimensional relationships, solve geometric problems, and understand how shapes occupy spaces in more complex, real-world ways.