Problem 26 - Precalculus with Calculus Previews

Question

The given three points form a triangle. Determine which triangles are isosceles and which are right triangles. $$ (0,0,0),(3,6,-6),(2,1,2) $$

Step-by-Step Solution

Answer

The triangle is a right triangle, but not isosceles.

Step 1: Label the Points

Let's start by labeling the three given points: \( A = (0,0,0) \), \( B = (3,6,-6) \), and \( C = (2,1,2) \). We will use these labels to refer to the vertices of the triangle.

Step 2: Calculate the Distances Between Each Pair of Points

To check if the triangle is isosceles or right, we first calculate the distances between each pair of points.The distance between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by the formula:\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \]**Distance \( AB \):**\[ AB = \sqrt{(3-0)^2 + (6-0)^2 + (-6-0)^2} = \sqrt{9 + 36 + 36} = \sqrt{81} = 9 \]**Distance \( BC \):**\[ BC = \sqrt{(2-3)^2 + (1-6)^2 + (2+6)^2} = \sqrt{1 + 25 + 64} = \sqrt{90} \]**Distance \( AC \):**\[ AC = \sqrt{(2-0)^2 + (1-0)^2 + (2-0)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

Step 3: Check for an Isosceles Triangle

An isosceles triangle has at least two equal sides. Let's check if any two distances are equal:- \( AB = 9 \)- \( BC = \sqrt{90} \)- \( AC = 3 \)Since no two distances are equal, the triangle is not isosceles.

Step 4: Check for a Right Triangle

A triangle is right if one of its angles is 90 degrees, which can be confirmed by the Pythagorean theorem: the square of the longest side is equal to the sum of the squares of the other two sides.Calculate:\[ AB^2 = 81, \quad BC^2 = 90, \quad AC^2 = 9 \]Check if any of the squares of the sides follows the equation:- \( 81 = 90 + 9 ot\implies\) Right Triangle- However, rearranging shows \( 90 = 81 + 9 \to \text{confirm}. \)Since the sum of \( AB^2 \) and \( AC^2 \) equals \( BC^2 \), the triangle is a right triangle.

Key Concepts

Distance Formula in 3D Geometry

To calculate the distance between two points in 3D space, you use the distance formula. This is a fundamental concept in geometry. The distance formula helps find the length of the line segment between any two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in a three-dimensional space. The formula is: \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \] Here's an easy way to think about it:

• It's just an extension of the Pythagorean theorem, which you might already know from 2D spaces.
• The formula squares the differences of respective coordinates, adds them up, and takes the square root of the total.

Using this formula, distances between points A, B, and C were computed, leading us to important observations about the triangle involved.

Understanding Isosceles Triangles

An isosceles triangle is a special type of triangle that has at least two sides of equal length. This characteristic makes isosceles triangles easy to identify once you have calculated all side lengths. Key traits of isosceles triangles include:

• Two sides are of equal length. This property also implies that two angles opposite these sides are equal.
• It can be easily checked by comparing calculated side lengths, as seen in the previous solution, where none matched, meaning our triangle is not isosceles.

While comparing the triangle's side lengths in our problem, we found they were all different, confirming that the triangle was not isosceles. Remember, checking for side equality is crucial for identifying isosceles triangles.

Right Triangles and the Pythagorean Theorem

Right triangles, another fascinating type of triangle, have one right angle. This special angle measures 90 degrees. The Pythagorean theorem plays a key role in identifying right triangles. It asserts that in a right triangle:

• The square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.

In mathematical terms: \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides. In our solution, the longest side \( BC \) was confirmed as the hypotenuse because \( BC^2 = AB^2 + AC^2 \). Thus, the triangle formed by points A, B, and C is a right triangle. Understanding this theorem helps immensely in geometry, providing a direct way to establish the presence of a right angle in a triangle.