Problem 26 - Calculus an Applied Approach

Question

Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither of these. $$ (5,3,4),(7,1,3),(3,5,3) $$

Step-by-Step Solution

Answer

Firstly, find the lengths of the sides using the distance formula in three-dimensional space. Then apply the Pythagorean theorem to check if the triangle is a right triangle and compare the lengths of the sides to check if the triangle is an isosceles triangle. If it isn't either, then it's neither right angled nor isosceles.

Step 1: Find the lengths of the sides

Calculate the lengths of the sides AB, BC and CA using the distance formula between two points in three-dimensional space, which is \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \).

Step 2: Check for a right triangle

Test if the triangle is a right triangle by using the Pythagorean theorem. If the square of one side length is equal to the sum of squares of the two other side lengths, then it is a right triangle. That is, if \( (AB)^2 = (BC)^2 + (CA)^2 \) or \( (BC)^2 = (AB)^2 + (CA)^2 \) or \( (CA)^2 = (AB)^2 + (BC)^2 \), then it is a right triangle.

Step 3: Check for an isosceles triangle

Check if any two sides of the triangle are equal. If yes, then the triangle is an isosceles triangle. That is, if AB = BC, or BC = CA, or CA = AB, then it is an isosceles triangle.

Key Concepts

Right Triangle Conditions

Understanding the conditions to determine if a triangle is a right triangle is essential in geometry. A right triangle is defined by having one angle precisely equal to 90 degrees, forming a "right" angle. But how do we check for this geometrically using the side lengths?

The Pythagorean theorem is the key tool. It's expressed through the relationship: \[c^2 = a^2 + b^2\]where \(c\) is the hypotenuse (the longest side), and \(a\) and \(b\) are the other two sides.

• Check each combination: Instead of just one formula, you need to consider all permutations of sides as potential hypotenuses.
• If any of these combinations satisfy the squared equality, the triangle is a right triangle.
• Recall nature of the hypotenuse: If none of the sides, when squared, equals the sum of the squares of the other two sides, it's not a right triangle.

To sum up, confirming the presence of a right angle helps to categorize the type of triangle you're dealing with, thereby influencing how you might analyze further properties like area and perimeter.

Isosceles Triangle Properties

Isosceles triangles possess distinct characteristics defined by their equal sides. Essentially, a triangle is isosceles if it has at least two sides that are equal in length. This elementary yet profound property simplifies many geometrical proofs and problems.

When identifying an isosceles triangle:

• Use the triangle's vertices and apply the distance formula to calculate its sides.
• Compare the computed side lengths: The existence of a pair of equal side lengths confirms it's an isosceles triangle.
• Remember: It can also be equilateral, meaning all sides are equal, which is a special case of isosceles.

What's fascinating about isosceles triangles is their symmetry. This symmetry leads to equal angles opposite the equal sides, which can be useful in problem-solving and understanding reflections. This property makes isosceles triangles a favorite in many fields that involve geometrical design.

Distance Formula in 3D

The distance formula in three-dimensional space is a crucial mathematical tool, enabling you to find the distance between any two points given their coordinates. Unlike two-dimensional geometry where you just have \(x\) and \(y\), three-dimensional analyses add a \(z\) component. Here's the formula:
\[\text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]This formula is derived from the Pythagorean theorem, extended into three dimensions. Each component, \((x_2-x_1)\), \((y_2-y_1)\), \((z_2-z_1)\), corresponds to the difference in the values from one point to the other.

Using this formula is straightforward yet powerful:

• Subtract each pair of coordinates.
• Square each of these differences to nullify negative results and emphasize magnitude.
• Add up these squares and take the square root to find the exact distance.

This procedure is fundamental not only in pure mathematics but also in applied sciences, engineering, and computer graphics, among others. Mastering it enhances your ability to navigate through spatial problems with efficiency.