1. The Pythagorean Theorem can be used to find distances between two points on a grid. But what about finding distances in three-dimensional space? Consider a rectangular box, which has length l, width w, and height h. What is the distance from one corner to the opposite corner in terms of l, w, and h?

Question

1. The Pythagorean Theorem can be used to find distances between two points on a grid. But what about finding distances in three-dimensional space? Consider a rectangular box, which has length l, width w, and height h. What is the distance from one corner to the opposite corner in terms of l, w, and h?

2. Find an example of an equilateral hexagon whose sides are all square root of 13 units long. Give coordinates for all six points. [Hint: Think carefully about where to draw the hexagon on a coordinate grid.]

Answer 1

If someone could please give me an explanation, I'd be very grateful. I have no idea how to solve number 1, and I feel like I'm over thinking number 2.

Answer 2

To find the distance between the opposite corners of a rectangular box in three-dimensional space, we can use the concept of the 3D Pythagorean Theorem.

The 3D Pythagorean Theorem states that in a right-angled triangle in three-dimensional space, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In the case of a rectangular box, the distance from one corner to the opposite corner forms the hypotenuse of a right-angled triangle. Let's call this distance D.

To find D, we can use the formula:

D = sqrt(l^2 + w^2 + h^2)

Here, l represents the length of the box, w represents the width, and h represents the height.

So, in terms of l, w, and h, the distance from one corner to the opposite corner in a rectangular box is given by the formula:

D = sqrt(l^2 + w^2 + h^2)

Now, let's consider the equilateral hexagon with sides of length sqrt(13) units.

The first step is to figure out where to draw the hexagon on a coordinate grid. To create an equilateral hexagon, we need to connect six points that are equidistant from each other.

Let's start by drawing a regular hexagon with one of its corners at the origin (0,0). Since all sides of the hexagon have a length of sqrt(13), we can determine the coordinates of the six points as follows:

Point 1: (0, 0)
Point 2: (sqrt(13), 0)
Point 3: (sqrt(13)/2, sqrt(3)*sqrt(13)/2)
Point 4: (-sqrt(13)/2, sqrt(3)*sqrt(13)/2)
Point 5: (-sqrt(13), 0)
Point 6: (-sqrt(13)/2, -sqrt(3)*sqrt(13)/2)

These coordinates form an equilateral hexagon on a coordinate grid where all the sides have a length of sqrt(13).