Problem 25 - College Algebra and 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. $$ (0,0,0),(2,2,1),(2,-4,4) $$

Step-by-Step Solution

Answer

The triangle is a right triangle. Side lengths are 3 for AB, 3√5 for BC, and 6 for AC. It is not an isosceles triangle as no two sides are of equal length.

Step 1: Calculate the Distance Between the Points

The distance between two points in 3D space is given by the equation \[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]. Use this equation to calculate the distance between the points (0,0,0) and (2,2,1), (2,2,1) and (2,-4,4), and (0,0,0) and (2,-4,4). These distances constitute the lengths of the sides of the triangle.

Step 2: Compute the Side Lengths

After performing the computations in Step 1, the side lengths of the triangle should be, \(AB=\sqrt{2^2 + 2^2 + 1^2}=3\), \(BC=\sqrt{0^2 + (-6)^2 + 3^2}=3\sqrt{5}\), and \(AC=\sqrt{2^2 + 4^2 + 4^2}=6\).

Step 3: Categorize the Triangle

Now, it is time to characterize the triangle. A right triangle satisfies the Pythagorean theorem \(AB^2 + BC^2 = AC^2\). An isosceles triangle has at least two equal sides. Given our side lengths, it is clear that the triangle is a right triangle because \(3^2 + (3\sqrt{5})^2 = 6^2\). However, none of the sides are equal length, so it is not an isosceles triangle.

Key Concepts

Distance Formula

In 3D geometry, to find the distance between two points, we use a specific formula that takes into account the differences in the x, y, and z coordinates. This formula is:\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]Where:

• \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the two points.
• The square root symbol \(\sqrt{}\) ensures that the distance is a non-negative number.

The formula measures the direct line distance between two points in space, known as the Euclidean distance. For example, using this formula, the distance between the points \((0,0,0)\) and \((2,2,1)\) is computed as:\[\sqrt{(2-0)^2 + (2-0)^2 + (1-0)^2} = \sqrt{2^2 + 2^2 + 1^2} = 3\]
This computation can be repeated for the additional sides of a triangle formed by points in 3D space to find all side lengths.

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. The presence of a right angle offers a unique property, adhering to the Pythagorean theorem, which states:\[a^2 + b^2 = c^2\]Here:

• \(a\) and \(b\) are the lengths of the shorter sides of the triangle, known as "legs."
• \(c\) is the length of the longest side, known as the "hypotenuse."

For instance, in our given triangle with sides \(3\), \(3\sqrt{5}\), and \(6\), we can check the Pythagorean theorem. Calculating:\[3^2 + (3\sqrt{5})^2 = 6^2\]\[9 + 45 = 36\]\[54 = 54\]Thus, it confirms the triangle is a right triangle. Calculating the square sums ensures precision when verifying whether the triangle includes a perfect right angle.
This method can also help in various applications where determining angles and relationships between lines is crucial.

Isosceles Triangle

An isosceles triangle is characterized by having at least two sides of equal length. This property means that it also has two equal angles opposite those sides. Recognizing an isosceles triangle is important in geometry due to its symmetry properties, which can simplify many mathematical problems.

• An isosceles triangle may resemble an equilateral triangle, where all sides are equal, but it only requires two equal sides.
• If a triangle does not have at least two equal side lengths, it cannot be classified as isosceles.

In our example, none of the sides—\(3\), \(3\sqrt{5}\), and \(6\)—are of the same length. Therefore, this triangle **does not** meet the criteria to be classified as an isosceles triangle. Checking for equal sides is straightforward and essential when identifying and understanding the types of triangles you are working with.