1. **Stating the problem:** We have a tetrahedron with edges labeled in terms of $a$ and constants. We need to find the value of $a$ given the edge lengths. 2. **Understanding the figure and given lengths:** The edges from the base to the apex are $2\sqrt{a}$. The base has a side length of 4, and an internal segment labeled $2\sqrt{3}$. Another internal edge is labeled 2. 3. **Using the Pythagorean theorem:** Since the apex edges are $2\sqrt{a}$ and the base side is 4, we can consider a right triangle formed by half the base side, the height from the apex to the base, and the edge $2\sqrt{a}$. 4. **Calculate half the base side:** Half of 4 is 2. 5. **Set up the Pythagorean relation:** Let the height from the apex to the base be $h$. Then, $$ (2\sqrt{a})^2 = h^2 + 2^2 $$ 6. **Simplify:** $$ 4a = h^2 + 4 $$ 7. **Use the internal segment $2\sqrt{3}$ to find $h$:** This segment likely represents the height $h$ or relates to it. Assuming $h = 2\sqrt{3}$, 8. **Substitute $h$ into the equation:** $$ 4a = (2\sqrt{3})^2 + 4 = 4 \times 3 + 4 = 12 + 4 = 16 $$ 9. **Solve for $a$:** $$ a = \frac{16}{4} = 4 $$ **Final answer:** $$ a = 4 $$