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.]

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.

actually the Pythagorean theorem is only used to find the length of a missing side of a right triangle, you use the distance formula which is

square root (x-x)^2+(y-y)^2

What if this is a three dimensional rectangle? The corners could be two corners in different regions.

For example, what if I had to measure the front bottom left corner to the rear top right corner?

1. To find the distance from one corner to the opposite corner of a rectangular box in three-dimensional space, you can use the 3D version of the Pythagorean Theorem. This theorem relates the lengths of the three sides of a right-angled triangle in three dimensions.

In the case of a rectangular box, the three sides of the triangle are the lengths of the three edges meeting at the corner where you want to find the distance. Let's call these sides l, w, and h, representing the lengths of the box's length, width, and height, respectively.

To find the distance from one corner to the opposite corner, you need to calculate the length of the hypotenuse of the right-angled triangle formed by the three sides. The formula for finding the length of the hypotenuse in three dimensions is:

d = √(l² + w² + h²)

Where d is the distance from one corner to the opposite corner of the rectangular box.

So, to find the distance, you square each of the three sides (l, w, and h), add them together, and then take the square root of the result. This will give you the distance between the two corners.

2. To find an example of an equilateral hexagon whose sides are all square root of 13 units long, you can start by considering the properties of an equilateral hexagon.

An equilateral hexagon is a polygon with six equal sides and six equal angles. To construct such a hexagon, you need to ensure that all the sides are the same length as the square root of 13 units.

Now, let's think about where to draw this hexagon on a coordinate grid. Since the sides are all the same length, and the angles are equal, the hexagon will be symmetric around its center.

To find the coordinates of the six points of the hexagon, you can start by choosing a central point. Let's say this point is at the origin of the coordinate system (0,0).

From the central point, you can draw three lines of the same length (square root of 13 units) at 120-degree angles to each other. The endpoints of these lines will be the coordinates of three points on the hexagon.

For example, choose a point at (0, square root of 13) as one of the points on the hexagon. This point is directly above the central point, and its y-coordinate is the square root of 13.

Now, to find the coordinates of the other two points, you can rotate this point around the central point by 120 degrees twice.

You can use rotation formulas or trigonometry to calculate the new coordinates. One way to do this is to use the fact that a 120-degree rotation around the origin corresponds to a rotation by 2π/3 radians.

So, the coordinates of the three points on the hexagon could be:

Point 1: (0, square root of 13)
Point 2: (square root of 13 * cos(2π/3), square root of 13 * sin(2π/3))
Point 3: (square root of 13 * cos(4π/3), square root of 13 * sin(4π/3))

These coordinates will give you an example of an equilateral hexagon with sides of length square root of 13 units.