To solve this problem, it looks like we are dealing with a right triangle, where:

• $a = 3\sqrt{2}$: one leg of the triangle,
• $c = 4\sqrt{5}$: the hypotenuse of the triangle.

The task is to find the length of the missing side $b$, which is the other leg.

We can use the Pythagorean Theorem, which states:

$a^2 + b^2 = c^2$

Step 1: Substitute the given values

Substitute $a = 3\sqrt{2}$ and $c = 4\sqrt{5}$ into the equation:

$(3\sqrt{2})^2 + b^2 = (4\sqrt{5})^2$

Step 2: Simplify the squares

1. Square $3\sqrt{2}$: $(3\sqrt{2})^2 = 9 \cdot 2 = 18$

2. Square $4\sqrt{5}$: $(4\sqrt{5})^2 = 16 \cdot 5 = 80$

Substitute these values back into the equation: $18 + b^2 = 80$

Step 3: Solve for $b^2$

$b^2 = 80 - 18$ $b^2 = 62$

Step 4: Solve for $b$

Take the square root of both sides: $b = \sqrt{62}$

The length of the missing side is: $\boxed{\sqrt{62}}$

Would you like a more detailed explanation or help with a related problem?

Related Questions:

1. How do you simplify square roots like $\sqrt{62}$?
2. What is the approximate decimal value of $\sqrt{62}$?
3. Can the Pythagorean Theorem apply to non-right triangles? Why or why not?
4. What are other real-life applications of the Pythagorean Theorem?
5. How do you check if three side lengths form a right triangle?

Tip:

Always check if the given sides are consistent with a right triangle (i.e., the hypotenuse should be the largest value) before applying the Pythagorean Theorem.