Question

pythagorean theorem (rounding)
score: 4/5 penalty: 1 off
question
find the length of the third side. if necessary, round to the nearest tenth.
image of a right triangle with legs 16 and 18? wait, no, the right angle is at the bottom, one leg is 16, the hypotenuse? wait, no, the triangle has one side labeled 16 (vertical leg), one side labeled 18 (the other leg? wait, no, the right angle is at the bottom, so the two legs are 16 and the unknown, and the hypotenuse is 18? wait, no, the image shows a right triangle with one leg 16, the other leg? wait, the labels: 16 on one leg, 18 on the other leg? wait, no, the right angle is at the bottom, so the vertical leg is 16, the horizontal leg is unknown, and the hypotenuse is 18? wait, no, the users image: the triangle has a right angle, one leg is 16 (vertical), the other leg (horizontal) is unknown, and the hypotenuse is 18? wait, no, maybe 18 is the hypotenuse. wait, the text: \find the length of the third side. if necessary, round to the nearest tenth.\ the triangle is a right triangle with legs 16 and the unknown, hypotenuse 18? or is 18 a leg? wait, the image: the right angle is at the bottom, so the two legs are 16 (vertical) and the horizontal leg (unknown), and the hypotenuse is 18 (the slant side). so we need to find the horizontal leg. wait, no, maybe 18 is a leg. wait, lets re-express: the triangle is right-angled, so pythagorean theorem: ( a^2 + b^2 = c^2 ), where ( c ) is hypotenuse. so if 16 and 18 are the two legs, then hypotenuse is ( sqrt{16^2 + 18^2} ). if one is a leg and the other is hypotenuse, then the third side is ( sqrt{18^2 - 16^2} ). wait, the image: the right angle is at the bottom, so the vertical leg is 16, the horizontal leg is unknown, and the hypotenuse is 18? wait, no, the slant side is labeled 18, and the vertical leg is 16. so hypotenuse is 18, one leg is 16, find the other leg. so ( a^2 + 16^2 = 18^2 ), so ( a^2 = 18^2 - 16^2 = 324 - 256 = 68 ), so ( a = sqrt{68} approx 8.2 ). wait, but maybe 18 is a leg. wait, the users image: the triangle has a right angle, one leg is 16 (vertical), another leg is 18 (horizontal), and hypotenuse is unknown. then hypotenuse is ( sqrt{16^2 + 18^2} = sqrt{256 + 324} = sqrt{580} approx 24.1 ). wait, the problem says \find the length of the third side\. so we need to check which is the hypotenuse. the right angle is at the bottom, so the two legs are the vertical and horizontal, and the hypotenuse is the slant. so if the slant is 18, then the third side (horizontal leg) is ( sqrt{18^2 - 16^2} ). if the slant is unknown, and the two legs are 16 and 18, then hypotenuse is ( sqrt{16^2 + 18^2} ). wait, the image: the labels are 16 (vertical leg) and 18 (the slant side, hypotenuse). so the third side is the horizontal leg. so ( a^2 + 16^2 = 18^2 ), so ( a = sqrt{18^2 - 16^2} = sqrt{324 - 256} = sqrt{68} approx 8.2 ). or if 18 is a leg, then hypotenuse is ( sqrt{16^2 + 18^2} approx 24.1 ). wait, the problem says \third side\, so we need to see which is given. the image: the right angle is at the bottom, so the two legs are 16 (vertical) and the horizontal (unknown), and the hypotenuse is 18 (slant). so the third side is the horizontal leg. so the calculation is ( sqrt{18^2 - 16^2} ).

Explanation:

Step1: Identify triangle type and sides

It is a right triangle with hypotenuse 18 and one leg 16. Let the third side be $x$.

Step2: Apply Pythagorean theorem

$$16^2 + x^2 = 18^2$$

Step3: Calculate squares

$$256 + x^2 = 324$$

Step4: Solve for $x^2$

$$x^2 = 324 - 256 = 68$$

Step5: Find $x$ and round

$$x = \sqrt{68} \approx 8.2$$

Answer:

8.2