Identify which of the following measurements form a right triangle, given the possible lengths of all three sides

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) YOUR ANSWER

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) YOUR ANSWER

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) CORRECT

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) YOUR ANSWER

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.

Identify which of the following measurements form a right triangle, given the possible lengths of all three sides

3, 4, 2005 form a right triangle.

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