Question
A cube. The top face has points C, A, B, D and the bottom face has points G, E, F, H. Diagonals are drawn from B to C and from C to F. Side B F is 11 feet.
What are the exact lengths of BC and CF in this cube?
The length, in feet, of BC is the square root of
.
The length, in feet, of CF is the square root of
.
What are the exact lengths of BC and CF in this cube?
The length, in feet, of BC is the square root of
.
The length, in feet, of CF is the square root of
.
Answers
GPT-4o mini
To solve for the lengths of \( BC \) and \( CF \) in the cube with vertices labeled as you described, let's assume the cube has a side length of \( s \).
### 1. Finding Length of \( BC \)
Points \( B \) and \( C \) are on the top face of the cube. The coordinates of these points can be defined as follows assuming:
- Point \( A \) is at \( (0, 0, s) \)
- Point \( B \) is at \( (s, 0, s) \)
- Point \( C \) is at \( (s, s, s) \)
- Point \( D \) is at \( (0, s, s) \)
Thus, the coordinates for the points \( B \) and \( C \) are:
- \( B = (s, 0, s) \)
- \( C = (s, s, s) \)
The length of \( BC \) can be calculated using the distance formula. Since \( B \) and \( C \) share the same \( x \) and \( z \) coordinates, we only need to consider the difference in the \( y \) coordinate:
\[
BC = |y_C - y_B| = |s - 0| = s
\]
### 2. Finding Length of \( CF \)
Now we find \( CF \). Points \( C \) and \( F \) are defined as follows. Since \( F \) lies on the bottom face of the cube, we have:
- Point \( F \) is at \( (s, s, 0) \)
Now, our points \( C \) and \( F \) have the coordinates:
- \( C = (s, s, s) \)
- \( F = (s, s, 0) \)
Again, we use the distance formula. Since \( C \) and \( F \) share the same \( x \) and \( y \) coordinates, we only need to consider the difference in the \( z \) coordinate:
\[
CF = |z_C - z_F| = |s - 0| = s
\]
### Conclusion
Since it is given that \( BF = 11 \) feet, we can deduce that the side length of the cube \( s \) must also be \( 11 \) feet, as \( BF \) spans one side length vertically.
Thus, the lengths are:
- The length of \( BC \) is the square root of \( s^2 = 11^2 = 121 \).
- The length of \( CF \) is the square root of \( s^2 = 11^2 = 121 \).
Therefore:
- The length, in feet, of \( BC \) is the square root of \( 121 \).
- The length, in feet, of \( CF \) is the square root of \( 121 \).
### 1. Finding Length of \( BC \)
Points \( B \) and \( C \) are on the top face of the cube. The coordinates of these points can be defined as follows assuming:
- Point \( A \) is at \( (0, 0, s) \)
- Point \( B \) is at \( (s, 0, s) \)
- Point \( C \) is at \( (s, s, s) \)
- Point \( D \) is at \( (0, s, s) \)
Thus, the coordinates for the points \( B \) and \( C \) are:
- \( B = (s, 0, s) \)
- \( C = (s, s, s) \)
The length of \( BC \) can be calculated using the distance formula. Since \( B \) and \( C \) share the same \( x \) and \( z \) coordinates, we only need to consider the difference in the \( y \) coordinate:
\[
BC = |y_C - y_B| = |s - 0| = s
\]
### 2. Finding Length of \( CF \)
Now we find \( CF \). Points \( C \) and \( F \) are defined as follows. Since \( F \) lies on the bottom face of the cube, we have:
- Point \( F \) is at \( (s, s, 0) \)
Now, our points \( C \) and \( F \) have the coordinates:
- \( C = (s, s, s) \)
- \( F = (s, s, 0) \)
Again, we use the distance formula. Since \( C \) and \( F \) share the same \( x \) and \( y \) coordinates, we only need to consider the difference in the \( z \) coordinate:
\[
CF = |z_C - z_F| = |s - 0| = s
\]
### Conclusion
Since it is given that \( BF = 11 \) feet, we can deduce that the side length of the cube \( s \) must also be \( 11 \) feet, as \( BF \) spans one side length vertically.
Thus, the lengths are:
- The length of \( BC \) is the square root of \( s^2 = 11^2 = 121 \).
- The length of \( CF \) is the square root of \( s^2 = 11^2 = 121 \).
Therefore:
- The length, in feet, of \( BC \) is the square root of \( 121 \).
- The length, in feet, of \( CF \) is the square root of \( 121 \).