Exploring the Unique Right-Angled Triangle with a 11-Meter High Leg

When faced with a mathematical challenge involving the geometry of a right-angled triangle, one might ask: if the height of a right-angled triangle is 11 meters and the base and hypotenuse are positive integers, what is the perimeter of the triangle? This question can be approached using the properties of Pythagorean triples, a set of three positive integers a, b, and c that can form the sides of a right-angled triangle, where a^2 b^2 c^2.

Understanding Pythagorean Triples and Their Properties

For any odd prime number ( P ), there is only one Pythagorean triple: ( (P, frac{P^2 - 1}{2}, frac{P^2 1}{2}) ).

Let's examine a few examples based on this rule: For ( 11 ): the Pythagorean triple is ( (11, frac{121 - 1}{2} 60, frac{121 1}{2} 61) ) For ( 7 ): the Pythagorean triple is ( (7, frac{49 - 1}{2} 24, frac{49 1}{2} 25) ) For ( 5 ): the Pythagorean triple is ( (5, frac{25 - 1}{2} 12, frac{25 1}{2} 13) ) For ( 3 ): the Pythagorean triple is ( (3, frac{9 - 1}{2} 4, frac{9 1}{2} 5) ) For ( 13 ): the Pythagorean triple is ( (13, frac{169 - 1}{2} 84, frac{169 1}{2} 85) )

Brute Force Approach for the Given Problem

In order to solve the problem described, a simple yet effective brute force method was employed. Using JavaScript, the triangle's sides were calculated for a height of 11 meters, with the base embedded within a loop that counts from 1 to 999. This method revealed that the only integer base that yields an integral solution is 60.

Calculation Details

The calculation for the sides of the right-angled triangle with a height of 11 meters and a base of 60 can be performed as follows:

112 602 121 3600 3721

Testing the square root of 3721 provides:

√3721 61, thus the hypotenuse is 61 meters.

Therefore, the triangle's sides are: 11, 60, and 61 meters. The perimeter of the triangle is the sum of all three sides:

11 60 61 132 meters

This concludes that the only integer solution for the given problem where the height is 11 meters and the sides are positive integers is the 11-60-61 triangle.

Uniqueness of the Solution

This solution is unique because for a leg of length 11, only one integer hypotenuse satisfies the equation. Testing all potential values for the other leg from 1 to 60 confirmed that there is no other solution. As the other leg must be greater than 60 and the hypotenuse must be greater than 61, any other solution would have the difference between the hypotenuse squared and the other leg squared greater than 112, thus ruling out any other combinations.

Conclusion

The perimeter of a right-angled triangle with one leg of 11 meters, and the base and hypotenuse as positive integers is 132 meters. This problem exemplifies how mathematical principles such as Pythagorean triples can be applied to find unique solutions.

Key Takeaways

The height of 11 meters corresponds to the Pythagorean triple 11, 60, 61. The perimeter of the triangle is 132 meters. No other integer solutions exist for a right-angled triangle with a leg of 11 meters and positive integer sides.

The exploration of the 11–60–61 triangle via a brute force method not only provides a concrete solution but also highlights the importance of understanding and applying mathematical principles to solve geometric problems.

Note: The only Pythagorean triple with a leg of 11 meters is 11-60-61, making the solution unique.