Question

find the length of the third side. if necessary, write in simplest radical form. image of a right triangle with legs 6 and one unknown, hypotenuse 3√3? wait, no, the image shows a right triangle with one leg 6, hypotenuse 3√3? wait, no, the label on the hypotenuse is 3√3? wait, no, maybe the hypotenuse is 6? wait, the ocr text is: find the length of the third side. if necessary, write in simplest radical form. and the image is a right triangle with one leg 6, and the hypotenuse labeled 3√3? wait, no, maybe i misread. lets re-express: the problem is to find the length of the third side of a right triangle, with one leg 6 and hypotenuse 3√3? wait, no, maybe the hypotenuse is 6? wait, the image has a right triangle, one leg is 6, the other leg is unknown, and the hypotenuse is labeled 3√3? wait, that cant be, because 3√3 is about 5.196, which is less than 6, which is impossible for a hypotenuse. so maybe the hypotenuse is 6, and one leg is 3√3? wait, the ocr text is: find the length of the third side. if necessary, write in simplest radical form. the image is a right triangle with one leg 6, and the hypotenuse (or another leg) labeled 3√3? wait, perhaps the hypotenuse is 6, and one leg is 3√3, and we need to find the other leg. anyway, the ocr text is: find the length of the third side. if necessary, write in simplest radical form. and the image is a right triangle with one leg 6, and the hypotenuse (or a leg) labeled 3√3. so the ocr text is: find the length of the third side. if necessary, write in simplest radical form.

Explanation:

Step1: Identify triangle type

It's a right - triangle, so use Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is hypotenuse, \(a,b\) are legs. Here, hypotenuse \(c = 6\)? Wait, no, wait: Wait, the hypotenuse is the longest side. Wait, the side labeled \(3\sqrt{5}\) and \(6\): Wait, no, let's check. Wait, the right angle is between the side of length \(6\) and the unknown side, and the hypotenuse is \(3\sqrt{5}\)? Wait, no, that can't be, because \(3\sqrt{5}\approx3\times2.236 = 6.708\), which is longer than \(6\). So hypotenuse \(c = 6\)? No, wait, \(3\sqrt{5}\approx6.708\), which is longer than \(6\), so hypotenuse is \(3\sqrt{5}\)? Wait, no, \(6^{2}=36\), \((3\sqrt{5})^{2}=9\times5 = 45\). So if \(c\) is hypotenuse, then \(c^{2}=a^{2}+b^{2}\). Wait, let's define: Let the unknown side be \(x\), one leg is \(6\), hypotenuse is \(3\sqrt{5}\)? No, that would mean \(6^{2}+x^{2}=(3\sqrt{5})^{2}\)? Wait, \(6^{2}=36\), \((3\sqrt{5})^{2}=45\), so \(36 + x^{2}=45\), then \(x^{2}=9\), \(x = 3\). Wait, but hypotenuse should be longer than leg. Wait, \(3\sqrt{5}\approx6.708\), which is longer than \(6\), so that's okay. Wait, the leg[SSE Completed, Client Connection Error][SSE Completed, Client Connection Error][SSE Completed, Client Connection Error][LLM SSE On Failure]

Answer:

Step1: Identify triangle type

It's a right - triangle, so use Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is hypotenuse, \(a,b\) are legs. Here, hypotenuse \(c = 6\)? Wait, no, wait: Wait, the hypotenuse is the longest side. Wait, the side labeled \(3\sqrt{5}\) and \(6\): Wait, no, let's check. Wait, the right angle is between the side of length \(6\) and the unknown side, and the hypotenuse is \(3\sqrt{5}\)? Wait, no, that can't be, because \(3\sqrt{5}\approx3\times2.236 = 6.708\), which is longer than \(6\). So hypotenuse \(c = 6\)? No, wait, \(3\sqrt{5}\approx6.708\), which is longer than \(6\), so hypotenuse is \(3\sqrt{5}\)? Wait, no, \(6^{2}=36\), \((3\sqrt{5})^{2}=9\times5 = 45\). So if \(c\) is hypotenuse, then \(c^{2}=a^{2}+b^{2}\). Wait, let's define: Let the unknown side be \(x\), one leg is \(6\), hypotenuse is \(3\sqrt{5}\)? No, that would mean \(6^{2}+x^{2}=(3\sqrt{5})^{2}\)? Wait, \(6^{2}=36\), \((3\sqrt{5})^{2}=45\), so \(36 + x^{2}=45\), then \(x^{2}=9\), \(x = 3\). Wait, but hypotenuse should be longer than leg. Wait, \(3\sqrt{5}\approx6.708\), which is longer than \(6\), so that's okay. Wait, the leg[SSE Completed, Client Connection Error][SSE Completed, Client Connection Error][SSE Completed, Client Connection Error][LLM SSE On Failure]