How to Determine the Legs of a Right Angle Triangle Using Only the Hypotenuse and Area

One of the most intriguing problems in geometry involves determining the lengths of the legs of a right angle triangle when only the hypotenuse and area are given. This article delves into the mathematical methods and formulas used to find the unknown side lengths.

Problem Statement and Equations

Suppose we have a right-angled triangle with legs matha/math and mathb/math, hypotenuse mathc/math, and area mathA/math. We aim to find matha/math and mathb/math given mathc/math and mathA/math.

Motivation and Key Equations

The problem can be approached using the Pythagorean theorem and the formula for the area of a triangle.

Pythagorean Theorem:

[a^2 b^2 c^2]

Area of the Triangle:

[A frac{1}{2}ab]

Combined Equations:

First, we solve for the sum and difference of the legs:

[a b sqrt{c^2 - 4A}]

[a - b sqrt{c^2 - 4A}]

Solving for matha/math and mathb/math

By adding and subtracting these equations, we can isolate matha/math and mathb/math.

[a frac{1}{2}(sqrt{c^2 - 4A} sqrt{c^2 - 4A})]

[b frac{1}{2}(sqrt{c^2 - 4A} - sqrt{c^2 - 4A})]

Example Solutions

Example 1: Hypotenuse c 25, Area A 150

Using the formulas derived:

[sqrt{c^2 - 4A} sqrt{25^2 - 4 times 150} sqrt{1225} 35]

[sqrt{c^2 - 4A} sqrt{25^2 - 4 times 150} sqrt{25} 5]

[a frac{1}{2}(35 5) 20]

[b frac{1}{2}(35 - 5) 15]

Verification:

[c sqrt{20^2 15^2} sqrt{625} 25]

[A frac{1}{2} times 20 times 15 150]

Example 2: Hypotenuse c 25, Area A 84

Using the formulas:

[sqrt{c^2 - 4A} sqrt{25^2 - 4 times 84} sqrt{961} 31]

[sqrt{c^2 - 4A} sqrt{25^2 - 4 times 84} sqrt{289} 17]

[a frac{1}{2}(31 17) 24]

[b frac{1}{2}(31 - 17) 7]

Verification:

[c sqrt{24^2 7^2} sqrt{625} 25]

[A frac{1}{2} times 24 times 7 84]

Alternative Solution: General Triangular Area Formula

The area formula for a right triangle can be generalized:

Let the sides be mathab/math and the hypotenuse mathc/math.

[text{Area} sqrt{frac{4a^2b^2 - a^2b^2 - c^2^2}{16}}]

Solving this quadratic equation:

[c^2 a^2b^2 pm 2sqrt{a^2b^2 - 4text{Area}^2}]

The sides matha/math and mathb/math are:

[a sqrt{frac{1}{2}c^2 mp sqrt{c^4 - 16text{Area}^2}}]

[b sqrt{frac{1}{2}c^2 pm sqrt{c^4 - 16text{Area}^2}}]

Application and Verifications

Let's check with the 3/4/5 triangle where mathc 5text{ and Area} 6/math.

[a sqrt{frac{1}{2}25 - sqrt{625 - 16 times 36}} sqrt{frac{1}{2}25 - sqrt{49}} sqrt{9} 3 quad text{checkmark}]

[b sqrt{frac{1}{2}25 sqrt{625 - 16 times 36}} sqrt{frac{1}{2}25 sqrt{49}} sqrt{16} 4 quad text{checkmark}]

Conclusion

By following these mathematical steps, we can efficiently determine the lengths of the legs of a right angle triangle using only the hypotenuse and area. This method is widely applicable in solving similar geometric problems.