To solve the problem, we use the Pythagorean theorem since the triangle contains a right angle.

Pythagorean theorem:

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

From the figure:

• $a = 3$ (one leg)
• $b = x$ (the height or the other leg)
• $c = 6$ (the hypotenuse)

Substitute these values into the Pythagorean theorem:

$3^2 + x^2 = 6^2$

Step 1: Simplify the equation

$9 + x^2 = 36$

Step 2: Solve for $x^2$

$x^2 = 36 - 9$ $x^2 = 27$

Step 3: Solve for $x$

$x = \sqrt{27}$ $x = \sqrt{9 \cdot 3}$ $x = 3\sqrt{3}$

Final Answer:

$x = 3\sqrt{3}$

If you have further questions or need additional steps clarified, let me know!

Related Questions:

1. How is the Pythagorean theorem derived?
2. Can we use trigonometry to solve this triangle? If yes, how?
3. How does the value of $x$ change if the hypotenuse doubles?
4. What is the significance of the square root simplification step?
5. Can this problem be solved graphically?

Tip:

Always ensure to simplify square roots to their simplest radical form for exact answers in mathematics.