Opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given below. List the coordinates of the other six vertices of the box.
(0, 0, 0); (4, 2, 2)
I can see the base as
(0,0,0), (4,0,0), (4,2,0) and (0,2,0)
it is also 2 units high.
Can you see the other 4 vertices?
To find the coordinates of the other six vertices of the rectangular box, we can use the given coordinates of the opposite vertices as a reference.
Let's label the given opposite vertices as follows:
Vertex A: (0, 0, 0)
Vertex B: (4, 2, 2)
To find the other vertices, we can use the following approach:
1. Find the coordinates for the other two vertices that have the same x-coordinate as A.
- We can find these vertices by keeping the x-coordinate the same as A and changing the y and z coordinates.
- The two vertices will have the same x-coordinate as A, so the x-coordinate for these vertices will be 0.
The coordinates for the two vertices with the same x-coordinate as A are:
Vertex C: (0, 2, 0)
Vertex D: (0, 2, 2)
2. Find the coordinates for the other two vertices that have the same y-coordinate as B.
- We can find these vertices by keeping the y-coordinate the same as B and changing the x and z coordinates.
- The two vertices will have the same y-coordinate as B, so the y-coordinate for these vertices will be 2.
The coordinates for the two vertices with the same y-coordinate as B are:
Vertex E: (4, 2, 0)
Vertex F: (4, 0, 0)
3. Find the coordinates for the other two vertices that have the same z-coordinate as B.
- We can find these vertices by keeping the z-coordinate the same as B and changing the x and y coordinates.
- The two vertices will have the same z-coordinate as B, so the z-coordinate for these vertices will be 2.
The coordinates for the two vertices with the same z-coordinate as B are:
Vertex G: (4, 0, 2)
Vertex H: (0, 0, 2)
Therefore, the coordinates of the other six vertices of the rectangular box are:
Vertex C: (0, 2, 0)
Vertex D: (0, 2, 2)
Vertex E: (4, 2, 0)
Vertex F: (4, 0, 0)
Vertex G: (4, 0, 2)
Vertex H: (0, 0, 2)
To find the coordinates of the other six vertices of the rectangular box, we need to consider that opposite vertices of a rectangular box are related to each other by the distances between them.
Let's denote the given opposite vertices as A(0, 0, 0) and B(4, 2, 2). To find the coordinates of the other six vertices, we can use the distance formula in three dimensions, which is:
Distance = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
We will find the distance between the given opposite vertices and use it to calculate the coordinates of the other vertices.
First, the length of the box can be obtained by finding the distance between the x-coordinates of the opposite vertices:
Length = √((4 - 0)^2) = √(16) = 4
Similarly, the width and height can be obtained by finding the distances between the corresponding y-coordinates and z-coordinates, respectively:
Width = √((2 - 0)^2) = √(4) = 2
Height = √((2 - 0)^2) = √(4) = 2
Now, using these dimensions, we can determine the coordinates of the other six vertices.
1. A(0, 0, 0)
2. B(4, 2, 2)
3. C(4, 0, 0) - This vertex is obtained by keeping the x-coordinate the same (4), while setting the y-coordinate (0) and z-coordinate (0).
4. D(0, 2, 0) - This vertex is obtained by keeping the y-coordinate the same (2), while setting the x-coordinate (0) and z-coordinate (0).
5. E(0, 0, 2) - This vertex is obtained by keeping the z-coordinate the same (2), while setting the x-coordinate (0) and y-coordinate (0).
6. F(4, 2, 0) - This vertex is obtained by keeping the y-coordinate the same (2), while setting the x-coordinate (4) and z-coordinate (0).
7. G(0, 2, 2) - This vertex is obtained by keeping the z-coordinate the same (2), while setting the x-coordinate (0) and y-coordinate (2).
8. H(4, 0, 2) - This vertex is obtained by keeping the x-coordinate the same (4), while setting the y-coordinate (0) and z-coordinate (2).
So, the coordinates of the other six vertices are:
C(4, 0, 0), D(0, 2, 0), E(0, 0, 2), F(4, 2, 0), G(0, 2, 2), and H(4, 0, 2).