This is a right triangle, and we can solve for $z$ using the Pythagorean theorem:

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

Here:

• One leg is 4,
• The other leg is 6,
• The hypotenuse is $z$.

Substitute the values:

$4^2 + 6^2 = z^2$

$16 + 36 = z^2$

$52 = z^2$

Take the square root:

$z = \sqrt{52}$

Simplify:

$z = \sqrt{4 \cdot 13} = 2\sqrt{13}$

Thus, $z = 2\sqrt{13}$.

Do you want further clarification or more examples?

1. What happens when you use Pythagorean theorem in other contexts?
2. How do you simplify square roots?
3. Can we verify this solution visually?
4. What are real-life applications of the Pythagorean theorem?
5. How would you use trigonometry to solve this triangle instead?

Tip: Always look for perfect squares when simplifying square roots!