Calculating the Area of a Right Triangle When Given the Hypotenuse and One Side

When dealing with a right triangle, if you know the length of the hypotenuse and one side, you can determine the area of the triangle using the Pythagorean theorem and the formula for the area of a triangle. This method is both straightforward and widely applicable in various fields, from basic geometry to more complex engineering applications. This article will walk you through the process step-by-step.

Introduction to the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:

a2 b2 c2

where c is the hypotenuse, and a and b are the lengths of the other two sides (also known as the legs).

Given Values and Calculations

Consider a right triangle where the hypotenuse c is 13 cm, and one of the other sides a is 5 cm. To find the area, we need to determine the length of the third side, b, which can be found using the Pythagorean theorem:

Start with the equation: c2 a2 b2 Substitute the given values: 132 52 b2 Calculate: 169 25 b2 Solve for b2: b2 169 - 25 144 Take the square root of both sides: b √144 12 cm

Calculating the Area

Once we have all three side lengths - 5 cm, 12 cm, and 13 cm - we can use the formula for the area of a right triangle:

Area 1/2 × base × height

Here, the base and height are the two legs of the right triangle, so:

Area 1/2 × 5 cm × 12 cm 30 cm2

Conclusion

In summary, by applying the Pythagorean theorem to find the missing side and then using the area formula for a right triangle, we can easily calculate the area given the hypotenuse and one side. This method is not only mathematically sound but also highly practical for solving real-world problems involving right triangles.

Further Exploration

Understanding how to use the Pythagorean theorem and the area formula widens your mathematical toolkit and can help in a variety of practical scenarios. Whether you're a student working on a geometry problem or an engineer tackling a complex design challenge, these skills are invaluable.

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