Question

find the length of the third side. if necessary, write in simplest radical form.
image of a right triangle with legs 7 and 8 (wait, no, one leg is 7, the other leg? wait, the right angle is between the side labeled 7 and the side labeled 8? wait, no, the triangle has a right angle, one leg is 7, the other leg? wait, the hypotenuse? wait, the sides: one leg is 7, the other leg? wait, the side labeled 8 is a leg? wait, the right triangle has legs 7 and 8? wait, no, the image shows a right triangle with one leg 7, another leg 8? wait, no, the labels: one side is 7 (vertical), one side is 8 (horizontal), and the hypotenuse is the third side. wait, the problem is to find the length of the third side (hypotenuse) of a right triangle with legs 7 and 8. wait, no, maybe one leg is 7, the other leg? wait, the image: right angle, one leg 7, one leg 8, hypotenuse? wait, no, the problem says \find the length of the third side. if necessary, write in simplest radical form.\ and the triangle has legs 7 and 8? wait, no, maybe one leg is 7, the other leg is 8? wait, no, the labels: 7 is one leg, 8 is the other leg? wait, no, the right angle is between 7 and 8? wait, the triangle: right angle, sides 7 (vertical), 8 (horizontal), so hypotenuse is sqrt(7² + 8²)? wait, no, maybe 7 is a leg, 8 is a leg, hypotenuse? wait, no, maybe 7 is a leg, 8 is the hypotenuse? wait, no, the image: the right angle is at the bottom left, one leg is 7 (down), one leg is 8 (right), and the hypotenuse is the third side. wait, no, the labels: 7 is one leg, 8 is the other leg? wait, no, the side labeled 8 is a leg, 7 is a leg, hypotenuse? wait, no, the problem is to find the third side. so its a right triangle, legs 7 and 8, find hypotenuse? wait, no, maybe 7 is a leg, 8 is the hypotenuse? wait, no, the image: the right angle is between 7 and the side labeled 8? wait, maybe the legs are 7 and x, hypotenuse 8? wait, no, the labels: 7 is a leg, 8 is a leg, hypotenuse is the third side. wait, the ocr text: the problem is to find the third side of a right triangle with legs 7 and 8? wait, no, maybe 7 is a leg, 8 is the hypotenuse? wait, no, the image shows a right triangle with one leg 7, another leg 8, hypotenuse? wait, no, the problem is to find the length of the third side. so if its a right triangle, then by pythagoras, third side (hypotenuse) is sqrt(7² + 8²) = sqrt(49 + 64) = sqrt(113). wait, but maybe 7 is a leg, 8 is the hypotenuse, then the other leg is sqrt(8² - 7²) = sqrt(64 - 49) = sqrt(15). wait, the image: lets check the ocr again. the image has a right triangle with one leg 7 (vertical), one leg 8 (horizontal), so hypotenuse is sqrt(7² + 8²) = sqrt(113). wait, but maybe the labels are different. anyway, the ocr text is: \find the length of the third side. if necessary, write in simplest radical form.\ and the image of a right triangle with sides 7 and 8 (legs) and hypotenuse as the third side. so the ocr text is: find the length of the third side. if necessary, write in simplest radical form.
image of right triangle with legs 7 and 8 (wait, no, one leg 7, one leg 8? or 7 is a leg, 8 is the hypotenuse? the image shows a right angle, one side 7 (vertical), one side 8 (horizontal), hypotenuse is the third side. so the ocr text is: find the length of the third side. if necessary, write in simplest radical form.
answer attempt 1 out of 2

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem, which states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\) (or if we are finding a leg, \(a^{2}=c^{2}-b^{2}\), but in this case, we are finding the hypotenuse since the two given sides are the legs). Here, \(a = 7\) and \(b = 8\).

Step2: Apply the Pythagorean theorem

Let the length of the third side (hypotenuse) be \(c\). Then \(c^{2}=7^{2}+8^{2}\). Calculate \(7^{2}=49\) and \(8^{2}=64\). So \(c^{2}=49 + 64=113\). Then \(c=\sqrt{113}\) (since the length of a side cannot be negative).

Answer:

\(\sqrt{113}\)