Problem 67 - University Calculus: Early Transcendentals

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 of the triangle is \( \sqrt{17} + \sqrt{33} + 6 \).

Step 1: Identify the formula for distance

To find the perimeter of the triangle, we need to calculate the sum of the lengths of its sides. The length of a side can be found using the distance formula. For two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in the 3D space, the distance \( d \) between them is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] We'll apply this formula to each pair of points.

Step 2: Calculate distance AB

Using the coordinates of points A and B, \( A(-1, 2, 1) \) and \( B(1, -1, 3) \), substitute into the distance formula:\[ AB = \sqrt{(1 - (-1))^2 + (-1 - 2)^2 + (3 - 1)^2} \]\[ AB = \sqrt{(2)^2 + (-3)^2 + (2)^2} \]\[ AB = \sqrt{4 + 9 + 4} \]\[ AB = \sqrt{17} \]

Step 3: Calculate distance BC

Using the coordinates of points B and C, \( B(1, -1, 3) \) and \( C(3, 4, 5) \), substitute into the distance formula:\[ BC = \sqrt{(3 - 1)^2 + (4 - (-1))^2 + (5 - 3)^2} \]\[ BC = \sqrt{(2)^2 + (5)^2 + (2)^2} \]\[ BC = \sqrt{4 + 25 + 4} \]\[ BC = \sqrt{33} \]

Step 4: Calculate distance CA

Using the coordinates of points C and A, \( C(3, 4, 5) \) and \( A(-1, 2, 1) \), substitute into the distance formula:\[ CA = \sqrt{(-1 - 3)^2 + (2 - 4)^2 + (1 - 5)^2} \]\[ CA = \sqrt{(-4)^2 + (-2)^2 + (-4)^2} \]\[ CA = \sqrt{16 + 4 + 16} \]\[ CA = \sqrt{36} \]\[ CA = 6 \]

Step 5: Calculate the perimeter of the triangle

Now that we have the lengths of all sides, we can find the perimeter by summing these lengths:\[ \text{Perimeter} = AB + BC + CA \]\[ \text{Perimeter} = \sqrt{17} + \sqrt{33} + 6 \]

Key Concepts

Perimeter of a Triangle

The perimeter is the total distance around a triangle, which in this case involves adding up the lengths of all its sides.
The notion of a perimeter applies to any polygon, but for a triangle, it simply means the sum of its three sides' lengths.
Calculating the perimeter of a triangle in a 3D space is similar to doing it in 2D, just requiring the 3D distance formula.

• First, identify the vertices of the triangle. Here, they are points A, B, and C with their respective coordinates.
• Next, calculate the distance between each pair of points: AB, BC, and CA, using the 3D distance formula.
• Finally, add these distances together to get the perimeter.

By understanding how the perimeter is structured, you'll be more successful in solving these problems.

Coordinate Geometry

Coordinate geometry, often called Cartesian geometry, is a way to define the location of points in a plane using pairs of numbers. These numbers, known as coordinates, provide a precise way to describe the exact position of any point.
In 3D or three-dimensional geometry, each point is defined by three coordinates, typically noted as

• \((x, y, z)\) - where each coordinate represents a measurement in one of the three dimensions.

This system allows mathematicians and engineers to perform calculations involving distances and areas, like finding the perimeter of a triangle in 3D space.
For our exercise, coordinates help us organize and calculate distances between each pair of vertices of the triangle.
Understanding coordinate geometry is crucial as it provides the foundation needed for calculating the distance formula.

3D Geometry

3D Geometry extends the concepts of 2D plane geometry into three dimensions, adding depth to the familiar length and width.
This field of geometry is crucial for understanding the real world, where objects aren't confined to a flat surface. In three dimensions, you can imagine moving up, down, left, right, forward, and backward.

• The 3D distance formula helps calculate the direct distance between two points in space, which includes the x, y, and z coordinates.
• To calculate it, you use \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\), where \(d\) is the distance between two points.

Using this formula helps us determine the sides of our triangle in any 3D coordinate system.
Since our triangle's vertices are points like in geometry, understanding these principles allows us to apply the necessary calculations to solve for the perimeter of the triangle in 3D space.