Question

find the length of the third side. if necessary, round to the nearest tenth.
right triangle with legs 9 and 13? wait, no, the right angle is at the vertex with sides 9 (one leg) and 13 (the other leg? wait, the image shows a right triangle with one leg 9, the other leg (horizontal) 13? wait, the ocr text: the triangle has a right angle, one leg 9, the other leg (lets say) 13? wait, the users image: the triangle has a right angle, one leg is 9, the other leg (the horizontal one) is 13? wait, the ocr text: the problem is to find the length of the third side (the hypotenuse or the other leg? wait, the right triangle: legs 9 and 13? wait, no, the image: the right angle is at the left vertex, one leg is vertical (length 9), the other leg is horizontal (length 13), and the hypotenuse is the third side. wait, the ocr text: \find the length of the third side. if necessary, round to the nearest tenth.\ then the triangle with right angle, one leg 9, the other leg 13? wait, maybe the horizontal leg is 13, vertical leg 9, so hypotenuse? wait, no, maybe the horizontal leg is 13, vertical leg 9, so the third side is the hypotenuse? wait, no, maybe the horizontal leg is 13, vertical leg 9, so the third side is the hypotenuse, or maybe one leg is 9, the hypotenuse is 13? wait, the image: the right angle is at the left, so the two legs are 9 (vertical) and 13 (horizontal), so the hypotenuse is the third side. wait, no, maybe the horizontal side is 13, vertical side is 9, so the third side (hypotenuse) is sqrt(9² + 13²)? wait, no, maybe the horizontal side is 13, and the vertical side is 9, so the hypotenuse is sqrt(9² + 13²) = sqrt(81 + 169) = sqrt(250) ≈ 15.8? wait, no, maybe the horizontal side is 13, and the other leg is 9, so the hypotenuse is sqrt(9² + 13²). wait, the ocr text: the problem is to find the length of the third side. so the triangle is right-angled, with two legs 9 and 13, so hypotenuse? or maybe one leg is 9, hypotenuse is 13, so the other leg is sqrt(13² - 9²) = sqrt(169 - 81) = sqrt(88) ≈ 9.4? wait, the image: the right angle is at the left, so the vertical leg is 9, the horizontal leg is 13, so the hypotenuse is the third side. wait, maybe the horizontal leg is 13, vertical leg 9, so hypotenuse is sqrt(9² + 13²) = sqrt(81 + 169) = sqrt(250) ≈ 15.8? wait, but the users image: the horizontal leg is labeled 13, vertical leg 9, so the third side is the hypotenuse. wait, the ocr text: the problem is to find the length of the third side. so the calculation is either pythagorean theorem: if its a right triangle, then c = sqrt(a² + b²) or b = sqrt(c² - a²). wait, the image: the right angle is at the left, so the two legs are 9 and 13, so hypotenuse is sqrt(9² + 13²) = sqrt(81 + 169) = sqrt(250) ≈ 15.8? wait, no, maybe the horizontal leg is 13, vertical leg 9, so the third side is the hypotenuse. wait, the ocr text: the problem is to find the length of the third side. so the ocr text is: \find the length of the third side. if necessary, round to the nearest tenth.\ then the triangle with right angle, one leg 9, the other leg 13 (horizontal), so hypotenuse. wait, maybe the horizontal leg is 13, vertical leg 9, so hypotenuse is sqrt(9² + 13²) = sqrt(250) ≈ 15.8? or maybe the horizontal leg is 13, and the hypotenuse is 13? no, the image: the horizontal leg is 13, vertical leg 9, right angle. so the third side is the hypotenuse. so the ocr text is: \find the length of the third side. if necessary, round to the nearest tenth.\ then the triangle with legs 9 and 13, so hypotenuse is sqrt(9² + 13²) = sqrt(81 + 169) = sqrt(250) ≈ 15.8. wait, but maybe the horizontal leg is 13, and the vertical leg is 9, so the third side is the hypotenuse. so the ocr text is as given: the problem is to find the third side of the right triangle with legs 9 and 13 (or maybe one leg 9, the other leg 13, so hypotenuse).

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\) (if we want to find the hypotenuse) or \(b^{2}=c^{2}-a^{2}\) (if we know the hypotenuse and one leg, and want to find the other leg). Here, we know one leg \(a = 9\) and the other leg (wait, no, wait: looking at the triangle, one leg is 9, the other leg is 13? Wait, no, the right - angle is between the leg of length 9 and the other leg, and the hypotenuse? Wait, no, in the diagram, the two legs are 9 and let's say \(x\), and the hypotenuse? Wait, no, the side labeled 13 is a leg? Wait, no, the right - angle is at the vertex where the sides of length 9 and the other leg meet, and the side of length 13 is a leg? Wait, no, I think I made a mistake. Wait, in a right - triangle, the hypotenuse is the longest side. Wait, 13 is longer than 9, so maybe 13 is the hypotenuse? Wait, no, if 9 and \(x\) are the legs, and 13 is the hypotenuse? Wait, no, 9 and 13: let's check. Wait, the Pythagorean theorem: if we have a right - triangle, \(c^{2}=a^{2}+b^{2}\), where \(c\) is the hypotenuse. Wait, maybe the two legs are 9 and \(x\), and the hypotenuse is 13? No, 9^{2}+x^{2}=13^{2}? Wait, 9^{2}=81, 13^{2}=169, then \(x^{2}=169 - 81=88\), \(x=\sqrt{88}\approx9.4\)? Wait, no, that can't be. Wait, maybe the leg is 9, the hypotenuse is 13? Wait, no, 9 is one leg, 13 is the other leg? No, the right - angle is between the leg of length 9 and the side of length 13? No, the diagram shows a right - triangle with one leg 9, another leg 13? Wait, no, I think I misread. Wait, the side labeled 13 is the hypotenuse? Wait, no, 9 and 13: let's recast. Let's assume that the two legs are \(a = 9\) and \(b\), and the hypotenuse is \(c\). Wait, no, the side with length 13 is a leg, and the side with length 9 is a leg, and we need to find the hypotenuse? Wait, that would be \(c=\sqrt{9^{2}+13^{2}}=\sqrt{81 + 169}=\sqrt{250}\approx15.8\). Wait, that makes more sense. Because 9 and 13 are the two legs, and we need to find the hypotenuse.

Step2: Apply the Pythagorean theorem

The Pythagorean theorem is \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 9\) and \(b = 13\).
First, calculate \(a^{2}\) and \(b^{2}\):
\(a^{2}=9^{2}=81\)
\(b^{2}=13^{2}=169\)
Then, \(a^{2}+b^{2}=81 + 169 = 250\)
Then, \(c=\sqrt{250}\). We can simplify \(\sqrt{250}=\sqrt{25\times10}=5\sqrt{10}\approx5\times3.1623\approx15.8\)

Answer:

\(15.8\)