Question

find the length of the third side. if necessary, write in simplest radical form.

Explanation:

Step1: Identify triangle type (right triangle)

This is a right triangle, so we use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a, b\) are the legs. Here, hypotenuse \(c = 5\sqrt{5}\), one leg \(a = 10\)? Wait, no—wait, the right angle is at the bottom, so the hypotenuse is the longest side. Wait, \(5\sqrt{5}\) is about \(5\times2.236 = 11.18\), and 10 is shorter. Wait, maybe I mixed up. Let's check: let the unknown side be \(x\), hypotenuse is \(5\sqrt{5}\)? No, wait, the side with length \(5\sqrt{5}\) is longer than 10? Wait, \(5\sqrt{5} \approx 11.18\), 10 is shorter. So actually, the hypotenuse is \(5\sqrt{5}\), and one leg is 10? No, that can't be, because hypotenuse must be longer than either leg. Wait, maybe the hypotenuse is \(5\sqrt{5}\)? No, 10 is 10, \(5\sqrt{5}\) is ~11.18, so 10 is a leg, \(5\sqrt{5}\) is the other leg? Wait, no, the right angle is at the bottom, so the two legs are the vertical and horizontal, and the hypotenuse is the slant. Wait, the given sides: one leg is 10, the other leg is \(x\), hypotenuse is \(5\sqrt{5}\)? No, that's impossible because \(5\sqrt{5} \approx 11.18\), but 10 is a leg, so hypotenuse must be longer than 10. Wait, maybe I got the hypotenuse wrong. Wait, the side labeled \(5\sqrt{5}\) is the hypotenuse? Wait, no, the right angle is at the bottom left, so the sides: one leg (horizontal) is \(x\), one leg (vertical) is 10, hypotenuse is \(5\sqrt{5}\)? No, that can't be, because \(5\sqrt{5}\) is about 11.18, which is longer than 10, but then \(x\) would be \(\sqrt{(5\sqrt{5})^2 - 10^2}\). Wait, let's compute that.

Step2: Apply Pythagorean theorem

Let’s denote the unknown leg as \(x\). The hypotenuse \(c = 5\sqrt{5}\), one leg \(a = 10\)? Wait, no—wait, maybe the hypotenuse is \(5\sqrt{5}\), and one leg is \(x\), the other leg is 10? Wait, no, hypotenuse must be longer than both legs. Wait, \(5\sqrt{5} \approx 11.18\), which is longer than 10, so 10 is a leg, \(x\) is the other leg, hypotenuse is \(5\sqrt{5}\)? Wait, no, that would mean \(x^2 + 10^2 = (5\sqrt{5})^2\). Let's compute:

\((5\sqrt{5})^2 = 25 \times 5 = 125\)

\(10^2 = 100\)

So \(x^2 = 125 - 100 = 25\)

Then \(x = \sqrt{25} = 5\)

Wait, that makes sense. So the unknown leg is 5.

Answer:

5