The problem involves finding the third side of a right triangle in its simplest radical form. The triangle's legs and hypotenuse are depicted on a coordinate plane.

Key Information:

• One leg extends vertically from $(1, 0)$ to $(1, 5)$, so its length is $5$.
• The hypotenuse connects $(1, 5)$ to $(3, 0)$ with a length of $15$.

We need to find the horizontal leg from $(1, 0)$ to $(3, 0)$. Let's calculate:

1. The vertical leg's length: $5$.
2. Use the Pythagorean theorem:

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

Where:

• $a = 5$,
• $c = 15$,
• $b$ is the side to find.

$5^2 + b^2 = 15^2$

$25 + b^2 = 225$

$b^2 = 200$

$b = \sqrt{200}$

Simplify $\sqrt{200}$: $\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}$

Final Answer:

The third side in simplest radical form is $10\sqrt{2}$.

Would you like a step-by-step breakdown or a graph for better visualization?

Related Questions:

1. How can you identify the hypotenuse in a right triangle?
2. What steps are involved in simplifying square roots?
3. Can you use the Pythagorean theorem with non-integer sides?
4. How does the coordinate plane help measure distances?
5. Why is $\sqrt{200}$ equivalent to $10\sqrt{2}$?

Tip:

Always simplify square roots by factoring out perfect squares.