Problem 7 - APEX Calculus

Question

The points \(A=(1,4,2), B=(2,6,3)\) and \(C=(4,3,1)\) form a triangle in space. Find the distances between each pair of points and determine if the triangle is a right triangle.

Step-by-Step Solution

Answer

The distances are \( \sqrt{6} \), \( \sqrt{17} \), and \( \sqrt{11} \); the triangle is a right triangle.

Step 1: Find the distance between points A and B

To find the distance between points A and B, use the distance formula:\[ \text{Distance AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]Substitute the coordinates of A (1,4,2) and B (2,6,3) into the formula:\[ \text{Distance AB} = \sqrt{(2 - 1)^2 + (6 - 4)^2 + (3 - 2)^2} = \sqrt{1^2 + 2^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]

Step 2: Find the distance between points B and C

Use the distance formula again to find the distance between B and C:\[ \text{Distance BC} = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2 + (z_3 - z_2)^2} \]Substitute the coordinates of B (2,6,3) and C (4,3,1):\[ \text{Distance BC} = \sqrt{(4 - 2)^2 + (3 - 6)^2 + (1 - 3)^2} = \sqrt{2^2 + (-3)^2 + (-2)^2} = \sqrt{4 + 9 + 4} = \sqrt{17} \]

Step 3: Find the distance between points C and A

Apply the distance formula for C and A:\[ \text{Distance CA} = \sqrt{(x_1 - x_3)^2 + (y_1 - y_3)^2 + (z_1 - z_3)^2} \]Substitute the coordinates of C (4,3,1) and A (1,4,2):\[ \text{Distance CA} = \sqrt{(1 - 4)^2 + (4 - 3)^2 + (2 - 1)^2} = \sqrt{(-3)^2 + 1^2 + 1^2} = \sqrt{9 + 1 + 1} = \sqrt{11} \]

Step 4: Check for a right triangle

A triangle is a right triangle if one of the squares of the largest sides equals the sum of the squares of the other two sides. Check the Pythagorean theorem:- Compare the squares: - Distance AB: \(6\) - Distance BC: \(17\) - Distance CA: \(11\) - Check combinations: - \(\sqrt{17}^2 = 17\) - \(\sqrt{6}^2 + \sqrt{11}^2 = 6 + 11 = 17\)- Since one combination works (\(\sqrt{BC} = \sqrt{AB}^2 + \sqrt{CA}^2 \)), the triangle is a right triangle.

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of its angles is exactly 90 degrees. This means it includes a right angle, hence the name "right triangle." An easy way to determine if a triangle is a right triangle is by using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is explained as:

• If a triangle has sides of length 'a,' 'b,' and 'c,' with 'c' being the largest, then it is a right triangle if -
  c^2 = a^2 + b^2.

In our problem, using 3D coordinates, we calculated the distances between each pair of points, known as the side lengths of the triangle, and verified one combination matched the Pythagorean theorem, proving it was a right triangle.

Triangle Geometry

Triangle geometry involves understanding the properties of triangles, including their angles, sides, and various other characteristics. In any triangle, the sum of the internal angles is 180 degrees.
Each triangle has three sides and three vertices. Depending on the angles and sides, triangles can be classified into different types, such as equilateral, isosceles, and scalene. Right triangles, as we examined, have that specific right angle.
Using geometry, you can explore different properties:

• Area: The space enclosed by the triangle, crucial for many geometric calculations.
• Perimeter: The sum of all the sides of the triangle.
• Centroid and Orthocenter: Points related to important intersections in the triangle, like medians and altitudes.

When analyzing a triangle through coordinates, such as in 3D space, geometry concepts help us conclude the type, area, and other specific details related to the triangle.

3D Coordinates

Working with 3D coordinates allows us to navigate and understand the spatial relations of points in three-dimensional space. In 3D geometry, each point is represented by an ordered triple a triplet (x, y, z) that indicates its location along the x, y, and z axes.
This system assists in visualizing problems that involve length, width, and depth, as in our triangle problem, where points A, B, and C each have their z component.
In 3D space:

• Distance Formula: Used to calculate the distance between two points by extending the 2D Pythagorean theorem to accommodate the z-coordinate: math \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \). It helps in determining side lengths.


• Visualization: Imagine placing the triangle in a 3D coordinate system, providing a clearer view of configurations.

Utilizing 3D coordinates is essential for complex problems across physics, engineering, and many applied mathematics fields, providing a comprehensive understanding beyond 2D systems.