The triangle in the image is a right triangle, and we are tasked with finding the height $x$, where one of the legs is $5$ and the hypotenuse of the right triangle is $20$.

We can solve for $x$ by applying the Pythagorean theorem, which is:

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

Where:

• $a = x$ (the height we are solving for),
• $b = 5$ (the given leg of the triangle),
• $c = 20$ (the hypotenuse).

Let's solve for $x$:

$x^2 + 5^2 = 20^2$ $x^2 + 25 = 400$ $x^2 = 400 - 25$ $x^2 = 375$ $x = \sqrt{375}$ $x = 5\sqrt{15}$

So, the value of $x$ is $5\sqrt{15}$, approximately equal to $19.36$.

Would you like a more detailed explanation or have any further questions?

Here are five related questions to expand on this:

1. How does the Pythagorean theorem apply to other triangles?
2. How would you solve for a missing side if you had an obtuse triangle?
3. What is the significance of the right angle in the Pythagorean theorem?
4. Can you calculate the other leg of the triangle if only the hypotenuse and $x$ were given?
5. What is the practical application of using $\sqrt{15}$ in exact form versus the decimal approximation?

Tip: Always remember that for right triangles, the hypotenuse is the longest side, and the Pythagorean theorem only applies to right-angled triangles.