Calculating the Perimeter of an Isosceles Right-Angled Triangle Given Its Area

Understanding the geometric properties of an isosceles right-angled triangle is crucial in various applications ranging from geometry and trigonometry to real-world problem-solving scenarios. This article will walk you through the steps to calculate the perimeter of such a triangle based on the area, a fundamental concept often encountered in geometry.

Understanding the Basics: Properties of an Isosceles Right-Angled Triangle

An isosceles right-angled triangle has two sides of equal length and a right angle (90 degrees) between these two sides. These properties make it easier to derive various geometric formulas related to its perimeter and area.

Step-by-Step Calculation

To find the perimeter of an isosceles right-angled triangle, you need to start with the formula for its area. The area (A) of an isosceles right-angled triangle can be expressed as:

[ A frac{1}{2} times b times h ]

Since the triangle is isosceles and right-angled, the base (b) and height (h) are equal. Let (x) be the length of each leg. Therefore, the area formula simplifies as:

[ A frac{1}{2} times x times x frac{x^2}{2} ]

Setting Up the Equation

We are given that the area (A) is 72 square centimeters. Substituting this value into the area formula, we get:

[ frac{x^2}{2} 72 ]

Multiplying both sides by 2 to isolate (x^2), we obtain:

[ x^2 144 ]

Solving for (x), we take the square root of 144:

[ x sqrt{144} 12 text{ cm} ]

Calculation of the Hypotenuse

To find the hypotenuse (c) of the triangle, we use the Pythagorean theorem:

[ c sqrt{x^2 x^2} sqrt{2x^2} xsqrt{2} 12sqrt{2} text{ cm} ]

Final Step: Calculation of the Perimeter

The perimeter (P) of the triangle is the sum of the lengths of all three sides. Therefore, the perimeter can be calculated as:

[ P x x c x x 12sqrt{2} 2x 12sqrt{2} ]

Substituting (x 12), we get:

[ P 2 times 12 12sqrt{2} 24 12sqrt{2} text{ cm} ]

For an approximate numerical value, considering ( sqrt{2} approx 1.414 ), the perimeter can be calculated as:

[ P approx 24 12 times 1.414 approx 24 16.968 approx 40.968 text{ cm} ]

Thus, the perimeter of the isosceles right-angled triangle is:

[ P 24 12sqrt{2} text{ cm} text{ or approximately } 40.97 text{ cm} ]

Conclusion

By following these steps, you can easily calculate the perimeter of an isosceles right-angled triangle given its area. This method not only helps in solving practical problems but also strengthens your understanding of geometric properties and theorems.