Question

question
find the length of the third side. if necessary, write in simplest radical form.
(right triangle with legs 5 and unknown, hypotenuse 10? wait, no, the triangle has a right angle, one leg 5, hypotenuse 10? wait, the image shows a right triangle with one leg 5, the other leg? wait, the side labeled 10 is the hypotenuse? wait, the ocr text: the triangle has a right angle, one leg 5, the other side (the hypotenuse?) labeled 10? wait, no, the users image: the triangle is right-angled, one leg is 5, the hypotenuse is 10? wait, no, maybe the legs are 5 and x, hypotenuse 10? wait, the ocr text: \find the length of the third side. if necessary, write in simplest radical form.\ and the image is a right triangle with one leg 5, the hypotenuse 10? wait, no, maybe the legs are 5 and the other leg, and the hypotenuse is 10? wait, the ocr text after extracting: \question
find the length of the third side. if necessary, write in simplest radical form.\ (and the image of a right triangle with one leg 5, the hypotenuse 10? or one leg 5, the other leg? wait, the number 10 is on the hypotenuse? wait, the users image: the triangle has a right angle, one leg 5, the hypotenuse 10? wait, no, maybe the legs are 5 and the other leg, and the hypotenuse is 10? 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 5, the side opposite? wait, maybe the legs are 5 and the other leg, and the hypotenuse is 10? wait, the ocr text is as above, with the image of a right triangle with one leg 5, the hypotenuse 10? or one leg 5, the other leg? wait, the key is the ocr text: the problem is to find the third side of a right triangle, with one leg 5, hypotenuse 10? or one leg 5, the other leg? wait, the image shows a right triangle with a right angle, one leg 5, the hypotenuse 10? wait, no, maybe the legs are 5 and the other leg, and the hypotenuse is 10? wait, the ocr text is: \find the length of the third side. if necessary, write in simplest radical form.\ and the image has a right triangle with one leg 5, the side labeled 10 (maybe hypotenuse). so the ocr text is: \question
find the length of the third side. if necessary, write in simplest radical form.\ (and the image of the right triangle with leg 5, hypotenuse 10? or leg 5, other leg? wait, the users image: the triangle has a right angle, one leg is 5, the hypotenuse is 10? or one leg 5, the other leg? wait, maybe the legs are 5 and the unknown, and the hypotenuse is 10? wait, no, in a right triangle, hypotenuse is the longest side. so if one leg is 5, and the hypotenuse is 10, then the other leg is sqrt(10^2 - 5^2) = sqrt(75) = 5*sqrt(3). but the ocr text is as above. so the ocr text is: \question
find the length of the third side. if necessary, write in simplest radical form.\

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. Let \(a = 5\), \(c=10\), and we need to find \(b\).
So, \(b^{2}=c^{2}-a^{2}\).

Step2: Substitute values

Substitute \(a = 5\) and \(c = 10\) into the formula:
\(b^{2}=10^{2}-5^{2}=100 - 25=75\).

Step3: Simplify the radical

\(b=\sqrt{75}=\sqrt{25\times3}=5\sqrt{3}\).

Answer:

\(5\sqrt{3}\)