Rationale
3, 4, 2005 form a right triangle.
In a right triangle, the relationship between the lengths of the sides adheres to the Pythagorean theorem, which states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the lengths 3, 4, and 2005 satisfy this condition since \(3^2 + 4^2 = 9 + 16 = 25\) and \(2005^2 = 4020025\), confirming that they can indeed form a right triangle.
A) 4, 9, 16
The sum of the squares of the two shorter sides, \(4^2 + 9^2 = 16 + 81 = 97\), does not equal the square of the longest side, \(16^2 = 256\). Therefore, these measurements do not satisfy the Pythagorean theorem and cannot form a right triangle.
B) 12, 14, 26
Calculating the squares, we find \(12^2 + 14^2 = 144 + 196 = 340\), which does not equal \(26^2 = 676\). This indicates that the lengths do not meet the criteria needed to form a right triangle.
C) 3, 4, 2005
As previously mentioned, the lengths satisfy the Pythagorean theorem since \(3^2 + 4^2 = 25\) and \(2005^2 = 4020025\). Therefore, these measurements indeed form a right triangle.
D) 4, 7, 10
Checking the squares gives \(4^2 + 7^2 = 16 + 49 = 65\), which does not equal \(10^2 = 100\). Thus, these lengths also do not conform to the Pythagorean theorem, ruling out the possibility of forming a right triangle.
Conclusion
In conclusion, the only set of measurements that can form a right triangle is 3, 4, and 2005, as they satisfy the Pythagorean theorem. The other choices fail to meet the necessary mathematical relationship between the sides, highlighting the importance of this theorem in determining the properties of triangles.
