Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that \((2,1,6),(4,7,9)\), and \((8,5,-6)\) are vertices of a right triangle. Hint: Only right triangles satisfy the Pythagorean Theorem.

Step-by-Step Solution

Answer

The points form a right triangle since they satisfy the Pythagorean theorem.

Step 1: Calculate the distance between the points

We'll use the distance formula to find the lengths of the sides of the triangle whose vertices are given. The distance between two points \(x_1, y_1, z_1\) and \(x_2, y_2, z_2\) is given by \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\.- For the distance between \(A(2,1,6)\) and \(B(4,7,9)\):\d_{AB} = \sqrt{(4 - 2)^2 + (7 - 1)^2 + (9 - 6)^2} = \sqrt{4 + 36 + 9} = \sqrt{49} = 7.\- For the distance between \(B(4,7,9)\) and \(C(8,5,-6)\):\d_{BC} = \sqrt{(8 - 4)^2 + (5 - 7)^2 + (-6 - 9)^2} = \sqrt{16 + 4 + 225} = \sqrt{245}\.- For the distance between \(A(2,1,6)\) and \(C(8,5,-6)\):\d_{AC} = \sqrt{(8 - 2)^2 + (5 - 1)^2 + (-6 - 6)^2} = \sqrt{36 + 16 + 144} = \sqrt{196} = 14.\Thus, the side lengths are 7, \(14\), and \(\sqrt{245}\).

Step 2: Use the Pythagorean Theorem

To check if the triangle is right, we verify if the sum of the squares of the two shorter sides equals the square of the longest side.- Calculate \(7^2 + (\sqrt{245})^2\) \implies 49 + 245 = 294.\- Calculate \(14^2\) \implies 196.\Since these two values are not equal, the intended pairs are incorrectly chosen. Let's re-evaluate by checking \( (14)^2 + (7)^2 = \sqrt{245}^2\).- Calculate \(14^2 + 7^2\) \implies 196 + 49 = 245\.- Calculate \(\sqrt{245}^2\) \implies 245\.These are equal, confirming that one angle is a right angle.

Key Concepts

Distance Formula

When you're working with points in 3D geometry, finding the distance between them is crucial. The **distance formula** is a mathematical equation that helps us calculate the straight-line distance between two points in a coordinate system. It's like a ruler that measures the straight line from one point to another in space. The formula you use is:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
This formula takes into account the differences in each axis point: the x-axis, the y-axis, and the z-axis.

To determine if points \((2,1,6),(4,7,9)\), and \((8,5,-6)\) form the vertices of a right triangle, calculate the distances (known as the side lengths) between them:

• Distance between points \(A\) and \(B\) is 7.
• Distance between points \(B\) and \(C\) is \(\sqrt{245}\).
• Distance between points \(A\) and \(C\) is 14.

Calculating these distances helps in understanding which are the potential triangle sides, crucial for the next step: applying the Pythagorean Theorem.

Pythagorean Theorem

The **Pythagorean Theorem** is a key concept that applies to right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is expressively simple:
\[c^2 = a^2 + b^2\]
Where:

• \(c\) is the hypotenuse.
• \(a\) and \(b\) are the other two sides.

In our exercise, to determine if the triangle formed by the calculated distances is a right triangle, we consider the longest side as the hypotenuse.
Thus, with side lengths of 7, 14, and \(\sqrt{245}\):
- Check if \(14^2 + 7^2 = (\sqrt{245})^2\).
After calculations:

• \(14^2 = 196\)
• \(7^2 = 49\)
• \(\sqrt{245}^2 = 245\)

Adding \(14^2 + 7^2\) gives 245, equal to \(\sqrt{245}^2\), confirming the triangle is indeed a right triangle.

3D Geometry

**3D Geometry** can be thought of as an extension of 2D methods into an extra dimension. In 3D geometry, every point has an additional coordinate—a 'z' coordinate, to represent depth, along with the 'x' and 'y' coordinates for length and height.

When working with points in 3 dimensions, concepts such as distance and angles require a bit more work compared to 2D geometry. It requires us to apply techniques like the distance formula over a new dimension. This is why we have to be cautious with our calculations, as the added dimension introduces more complexity.

For our exercise, using the correct concepts in 3D geometry was critical. With our points \((2,1,6),(4,7,9)\), and \((8,5,-6)\), understanding how to measure distances and apply the Pythagorean theorem helped to confirm the geometric nature of the triangle. These methods are foundational to navigating more complex geometric analyses in three-dimensional space.