Question

find the length of the third side. if necessary, write in simplest radical form.
image of a right triangle with legs √3 and 1, right angle between them

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle (has a right - angle symbol). Let the legs be \(a = 1\) and \(b=\sqrt{3}\), and the hypotenuse be \(c\) (wait, no, wait. Wait, in a right - triangle, the Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. Wait, but here we need to check which sides are given. Wait, the two legs? Wait, no, one leg is \(1\), one leg is \(\sqrt{3}\)? Wait, no, wait, maybe we need to find the other leg or the hypotenuse. Wait, let's assume that the two given sides are a leg and the hypotenuse? Wait, no, the right - angle is between the side of length \(1\) and the side we need to find? Wait, no, the given sides are \(\sqrt{3}\) and \(1\), with the right - angle between them? Wait, no, the right - angle is at the vertex where the side of length \(1\) and the unknown side meet. Wait, let's re - examine.

Wait, the right - triangle has one leg of length \(1\), one leg of length \(x\) (unknown), and the hypotenuse of length \(\sqrt{3}\)? No, wait, the side labeled \(\sqrt{3}\) and the side labeled \(1\): if the right - angle is between the side of length \(1\) and the unknown side, then the hypotenuse is \(\sqrt{3}\). Wait, no, let's use the Pythagorean theorem correctly. Let's denote the sides: let the legs be \(a = 1\) and \(b\) (unknown), and the hypotenuse \(c=\sqrt{3}\). Then by Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). So \(1^{2}+b^{2}=(\sqrt{3})^{2}\).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Let's assume that \(a = 1\), \(c=\sqrt{3}\), and we need to find \(b\). Then:

\(1^{2}+b^{2}=(\sqrt{3})^{2}\)

\(1 + b^{2}=3\)

Subtract \(1\) from both sides:

\(b^{2}=3 - 1=2\)

Take the square root of both sides:

\(b=\sqrt{2}\)? Wait, no, wait, maybe I got the sides wrong. Wait, maybe the two given sides are the legs, and we need to find the hypotenuse. Let's check: if \(a = 1\) and \(b=\sqrt{3}\) are the legs, then the hypotenuse \(c\) is given by \(c^{2}=1^{2}+(\sqrt{3})^{2}=1 + 3=4\), so \(c = 2\). Wait, that makes sense. Oh! I made a mistake earlier. The right - angle is between the two legs: one leg is \(1\), one leg is \(\sqrt{3}\), and we need to find the hypotenuse.

So, step 1: Identify the triangle as a right - triangle, so we can use the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), where \(a = 1\), \(b=\sqrt{3}\), and \(c\) is the hypotenuse (the third side we need to find).

Step 2: Substitute the values into the Pythagorean theorem.

\(c^{2}=1^{2}+(\sqrt{3})^{2}\)

\(c^{2}=1 + 3\)

\(c^{2}=4\)

Step 3: Take the square root of both sides. Since \(c\) is a length, we take the positive square root.

\(c=\sqrt{4}=2\)

Answer:

\(2\)