Determining the Legs of a Right Triangle Given Its Hypotenuse and Area

Understanding the relationship between the legs, hypotenuse, and area of a right triangle is a fundamental concept in geometry. This article will guide you through the process of determining the lengths of the legs of a right triangle given its hypotenuse and area. We will solve a specific example to illustrate the calculations.

Problem Statement

The hypotenuse of a right triangle is 15 m, and its area is 3 square meters. We need to find the lengths of the legs of the triangle.

Solution Approach

To solve this problem, we will use the following steps:

Use the formula for the area of a right triangle. Apply the Pythagorean theorem. Solve the resulting equations to find the lengths of the legs.

Step-by-Step Solution

Let's denote the lengths of the legs of the triangle as a and b. We know the following:

The area of the triangle is given by 3 square meters:

[frac{1}{2}ab 3]

This simplifies to:

[ab 6]

The hypotenuse is 15 meters:

Using the Pythagorean theorem:

[a^2 b^2 15^2]

This simplifies to:

[a^2 b^2 225]

Solving the Equations

We now have a system of two equations:

[ab 6]

[a^2 b^2 225]

To solve these equations, we can use the following steps:

1. Express b in terms of a from the area equation:

[b frac{6}{a}]

Substitute this into the Pythagorean equation:

[a^2 left(frac{6}{a}right)^2 225]

Simultaneously solve these equations to derive:

[a^2 frac{36}{a^2} 225]

Multiply through by a^2 to clear the denominator:

[a^4 36 225a^2]

Move all terms to one side to form a quadratic equation in a^2:

[a^4 - 225a^2 36 0]

Let x a^2, transforming the equation to:

[x^2 - 225x 36 0]

Solve this quadratic equation using the quadratic formula:

[x frac{-(-225) pm sqrt{(-225)^2 - 4 cdot 1 cdot 36}}{2 cdot 1}]

[x frac{225 pm sqrt{50625 - 144}}{2}]

[x frac{225 pm sqrt{50481}}{2}]

[x frac{225 pm 224.67}{2}]

Thus, we get two solutions for x (or a^2):

[x_1 frac{449.67}{2} 224.835]

[x_2 frac{0.33}{2} 0.165]

The valid physical solution (since it must be positive and the legs must be real numbers) is:

[a^2 224.835]

So:

[a sqrt{224.835} approx 15.0]

Substitute this back into the expression for b:

[b frac{6}{15} approx 0.40]

Therefore, the lengths of the legs of the right triangle are approximately:

[a approx 15.0 text{ m}]

[b approx 0.40 text{ m}]

Conclusion

The process of determining the lengths of the legs of a right triangle given its hypotenuse and area involves using the area formula and the Pythagorean theorem. By solving the resulting system of equations, we can find the exact or approximate lengths of the legs.

Additional Tips

When dealing with such problems, always check the physical constraints, such as ensuring that the values are real and positive. This ensures that the resulting triangle is valid.