Problem 61 - Thomas Calculus

Question

Find the perimeter of the triangle with vertices \(A(-1,2,1),\) \(B(1,-1,3),\) and \(C(3,4,5) .\)

Step-by-Step Solution

Answer

The perimeter is \(\sqrt{17} + \sqrt{33} + 6.\)

Step 1: Calculate Distance AB

To find the distance between points are required to use the distance formula:\(AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2},\)where \(A(-1,2,1)\) and \(B(1,-1,3)\). Therefore:\[AB = \sqrt{(1 - (-1))^2 + (-1 - 2)^2 + (3 - 1)^2} = \sqrt{(2)^2 + (-3)^2 + (2)^2} = \sqrt{4 + 9 + 4} = \sqrt{17}.\]

Step 2: Calculate Distance BC

Now calculate the distance between points \(B(1,-1,3)\) and \(C(3,4,5)\) using the distance formula mentioned in step 1:\[BC = \sqrt{(3 - 1)^2 + (4 - (-1))^2 + (5 - 3)^2} = \sqrt{(2)^2 + (5)^2 + (2)^2} = \sqrt{4 + 25 + 4} = \sqrt{33}.\]

Step 3: Calculate Distance AC

Calculate the distance between points \(A(-1,2,1)\) and \(C(3,4,5)\) using the distance formula:\[AC = \sqrt{(3 - (-1))^2 + (4 - 2)^2 + (5 - 1)^2} = \sqrt{(4)^2 + (2)^2 + (4)^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6.\]

Step 4: Sum all Sides to Find Perimeter

The perimeter of triangle \(ABC\) is the sum of the lengths of its sides: \[P = AB + BC + AC = \sqrt{17} + \sqrt{33} + 6.\]

Key Concepts

Perimeter of a Triangle

To find the perimeter of a triangle in 3D space, like the triangle with vertices \(A(-1,2,1)\), \(B(1,-1,3)\), and \(C(3,4,5)\), we must first calculate the distance between each pair of vertices. Each distance represents the length of one side of the triangle. Once we have the lengths for all sides, the perimeter is simply the sum of these lengths.

For the given triangle, the distances are calculated as follows:

• Distance \(AB = \sqrt{17}\)
• Distance \(BC = \sqrt{33}\)
• Distance \(AC = 6\)

Thus, the perimeter \(P\) is given by:\[P = AB + BC + AC = \sqrt{17} + \sqrt{33} + 6.\]

Understanding this concept is crucial as it allows you to determine the total boundary length of any triangular surface in 3D, which is essential for applications in engineering, architecture, and other fields that deal with spatial figures.

3D Geometry

3D geometry involves studying shapes in three-dimensional space, often called spatial geometry. Unlike 2D geometry, where shapes have only width and height, 3D geometry adds depth into the equation. This concept includes various figures such as spheres, cubes, prisms, and of course, triangular shapes like the one discussed in this problem.

In 3D, each point is represented by a combination of three values (x, y, z), which help determine its exact position in space. These coordinates allow us to navigate and calculate distances effectively between points.

Dealing with 3D geometry is key to understanding and solving real-world problems that involve volumes and spatial relationships. By grasping the basic principles of 3D shapes and distances, you can apply them to tasks such as modeling, design, and navigation across various domains.

Distance Between Points

Calculating distances between points in 3D geometry is a fundamental task, and it uses the distance formula. When working in a 3D space, each point \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is part of the calculation.

The distance \(d\) between two such points is calculated using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]This formula extends the Pythagorean theorem into three dimensions and allows us to measure how far apart two points are in space.

Understanding and using this formula is crucial for solving problems involving lengths, perimeters, and even surface areas in 3D. Whether you're planning routes, designing objects, or analyzing movement paths, the ability to calculate precise distances ensures accuracy in spatial assessments.