Final answer:
None of the options provided perfectly match the criteria for the sides of a 45°-45°-90° triangle, as the lengths do not adhere to the specific ratio expected of an isosceles right triangle. The lengths of the legs should be equal, and the hypotenuse should be √2 times longer than each leg.
Step-by-step explanation:
The student's question is: Which of the following could be the lengths of the sides of a 45°-45°-90° triangle? To solve this, we must recognize that in a 45°-45°-90° triangle, also known as an isosceles right triangle, the lengths of the legs (the sides opposite the 45° angles) are equal, and the hypotenuse (the side opposite the 90° angle) is √2 times longer than each leg. Therefore, if we denote the length of each leg as 'a', then the hypotenuse 'c' must be 'a√2'. Using the Pythagorean theorem, a² + a² = c², we can verify this relationship as 2a² = (a√2)².
Now let's evaluate the options given:
- (a) √3,2,√2 cannot be the sides of a 45°-45°-90° triangle because the two legs are not equal.
- (b) 3√2/2, 3√2,3 could potentially be the sides if we consider 3√2/2 as the length of the legs. However, by multiplying 3√2/2 by √2, we do not get 3, so this is not correct.
- (c) 13.32,13 could be the sides of such a triangle if we assume 13 to be the leg length, but when we multiply 13 by √2, we do not get exactly 13.32, though it is very close. Without further precision, we cannot confirm this.
- (d) 3,4,5 are the sides of a right triangle, but not an isosceles right triangle, so this set does not form a 45°-45°-90° triangle.
None of the options provided perfectly match the criteria for the sides of a 45°-45°-90° triangle. For such a triangle, we would expect to see a pair of equal leg lengths and a hypotenuse length that is the product of one leg length and √2.