Final answer:
The triangle with side lengths 10, 12, and 20 is not a valid triangle because the side lengths do not satisfy the triangle inequality theorem. Specifically, one side is longer than the sum of the other two, which is not possible in any triangle.
Step-by-step explanation:
When considering a triangle with side lengths, we recall that for it to be a specific type of triangle, such as a right triangle, the side lengths must satisfy the Pythagorean theorem. For a set of side lengths to represent a right triangle, the square of the length of the hypotenuse (the longest side) should equal the sum of the squares of the other two sides.
In this case, with side lengths of 10, 12, and 20, we can check if 102 + 122 equals 202. Calculating, we find that 102 + 122 is 100 + 144, which equals 244, and 202 is 400. Since 244 does not equal 400, this set of side lengths does not satisfy the Pythagorean theorem, which means this triangle is not a right triangle. Furthermore, since one side (20) is longer than the sum of the other two sides (10 + 12 = 22), this set of side lengths cannot form a triangle. A triangle inequality theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.