What is the volume of tetrahedron ABCD with edge lengths AB=2 AC=3 AD=4 BC=square root of 13 BD= 2 times square root of 5 and CD=5 ?

To find the volume of a tetrahedron with vertices \( A \), \( B \), \( C \), and \( D \) and given edge lengths, we can use the **Cayley-Menger determinant**.

The volume \( V \) of a tetrahedron with vertices \( A \), \( B \), \( C \), and \( D \) is given by the formula:

\[
V^2 = \frac{1}{288} \begin{vmatrix}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & AB^2 & AC^2 & AD^2 \\
1 & AB^2 & 0 & BC^2 & BD^2 \\
1 & AC^2 & BC^2 & 0 & CD^2 \\
1 & AD^2 & BD^2 & CD^2 & 0
\end{vmatrix}
\]

### Step 1: Insert the given edge lengths.
We know the following:

- \( AB = 2 \)
- \( AC = 3 \)
- \( AD = 4 \)
- \( BC = \sqrt{13} \)
- \( BD = 2\sqrt{5} \)
- \( CD = 5 \)

Square these lengths:

- \( AB^2 = 4 \)
- \( AC^2 = 9 \)
- \( AD^2 = 16 \)
- \( BC^2 = 13 \)
- \( BD^2 = 20 \)
- \( CD^2 = 25 \)

### Step 2: Construct the Cayley-Menger determinant.

\[
V^2 = \frac{1}{288} \begin{vmatrix}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & 4 & 9 & 16 \\
1 & 4 & 0 & 13 & 20 \\
1 & 9 & 13 & 0 & 25 \\
1 & 16 & 20 & 25 & 0
\end{vmatrix}
\]

### Step 3: Compute the determinant.

Now, I will calculate the determinant.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the remaining steps to compute the determinant by hand, or do you have another query?

Here are 5 related questions to expand on this topic:

1. What is the geometric interpretation of the Cayley-Menger determinant?
2. How can the volume formula for a tetrahedron be derived using vectors?
3. What are some other applications of determinants in geometry?
4. How can we generalize this method to higher-dimensional simplices?
5. What is the relationship between the volume of a tetrahedron and its circumscribed sphere?

**Tip:** For tetrahedrons, understanding the geometric properties of its edges can provide deeper insights into its spatial structure.

What is the volume of tetrahedron ABCD with edge lengths AB=2 AC=3 AD=4 BC=square root of 13 BD= 2 times square root of 5 and CD=5?

Explore how to calculate the volume of a tetrahedron with edge lengths AB=2, AC=3, AD=4, BC=√13, BD=2√5, and CD=5 using the Cayley-Menger determinant. This problem involves applying advanced geometry and linear algebra concepts, suitable for students at the college level or advanced high school geometry.

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