Final answer:
The sets (b) 5, 9, 10 and (d) 5, 12, 13 can create right triangles. The length of the other leg is 5 millimeters.
Step-by-step explanation:
The set of line segments that could create a right triangle are (b) 5, 9, 10 and (d) 5, 12, 13. To determine if a set of line segments could create a right triangle, we can use the Pythagorean theorem which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
For example, in the set 5, 9, 10, we have 5² + 9² = 25 + 81 = 106, which is not equal to 10² = 100, so this set does not form a right triangle.
However, in the set 5, 12, 13, we have 5² + 12² = 25 + 144 = 169, which is equal to 13² = 169, so this set does form a right triangle.
To find the length of the other leg of a right triangle when given the length of one leg and the length of the hypotenuse, we can use the Pythagorean theorem again. Let's call the length of the other leg x.
We have x² + 12² = 13², which simplifies to x² + 144 = 169. Subtracting 144 from both sides, we get x² = 25.
Taking the square root of both sides, we find that x = 5 or x = -5.
Since length cannot be negative, the length of the other leg of the right triangle is 5 millimeters.