Pythagorean Theorem in 3D

The Pythagorean Theorem states that the square of the length of the longest side (the hypotenuse) of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

Table of Contents:

• Pythagorean Theorem in 3D
• Solved Examples
• FAQs

Pythagorean Theorem in 3D

The Pythagorean Theorem can be extended to three-dimensional space, where it is called the Pythagorean Theorem in 3D. In 3D space, the theorem states that the square of the length of the longest side (the hypotenuse) of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

In mathematical notation, the Pythagorean Theorem in 3D can be written as:

c^2 = a^2 + b^2 + h^2

where c is the length of the hypotenuse, and a, b, and h are the lengths of the other three sides of the right triangle.

In 3D space, the sides of a right triangle are called legs, and the longest side (opposite the right angle between the height and diagonal of the base) is called the hypotenuse, just as in 2D space. The Pythagorean Theorem in 3D can be used to find the length of the hypotenuse or the length of one of the legs of a right triangle in 3D space.

The Pythagorean Theorem in 3D has many applications in fields such as physics, engineering, and computer graphics, where it is used to calculate distances, lengths, and angles in 3D space.

Solved Examples

Example 1:

Find the distance between two points in 3D space with coordinates (1, 2, 3) and (4, 5, 6).

Solution:

We can model this situation as a right triangle in 3D space, with the distance between the two points as the hypotenuse, and the differences in x, y, and z coordinates as the legs. Using the Pythagorean Theorem in 3D, we have:

c^2 = a^2 + b^2 + h^2

c^2 = (4 – 1)^2 + (5 – 2)^2 + (6 – 3)^2

c^2 = 3^2 + 3^2 + 3^2

c^2 = 27

c = sqrt(27)

Therefore, the distance between the two points is approximately 5.196 units.

Example 2:

A rectangular box has dimensions of 3 feet by 4 feet by 5 feet. What is the length of the longest diagonal of the box?

Solution:

We can model this situation as a right triangle in 3D space, with the diagonal of the box as the hypotenuse, and the dimensions of the box as the legs. Using the Pythagorean Theorem in 3D, we have:

c^2 = a^2 + b^2 + h^2

c^2 = 3^2 + 4^2 + 5^2

c^2 = 9 + 16 + 25

c^2 = 50

c = sqrt(50)

Therefore, the length of the longest diagonal of the box is approximately 7.071 feet.

FAQs

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