Calculating the Perimeter of a Right-Angled Isosceles Triangle with a Hypotenuse of 5m

In this article, we will explore the method to calculate the perimeter of a right-angled isosceles triangle when the hypotenuse is given to be 5 meters in length. This triangle has its base and height equal in length, allowing us to apply the Pythagorean Theorem for a precise calculation.

Understanding the Problem

Let the length of the base and height be x meters. Given the hypotenuse of the right-angled triangle is 5 meters, we can use the Pythagorean Theorem to find the value of x.

Step-by-Step Calculation

The Pythagorean Theorem states that in a right-angled triangle:

a^2 b^2 c^2

Where c is the hypotenuse and a and b are the sides (or legs) of the triangle. Since the base and height are equal, we have:

x^2 x^2 5^2

This simplifies to:

2x^2 25

Dividing both sides by 2:

x^2 25/2

Taking the square root of both sides:

x 5/√2

Now, that we have the value of x, we can find the perimeter of the triangle.

Calculating the Perimeter

The perimeter of a triangle is the sum of the lengths of its sides. For our right-angled isosceles triangle:

Perimeter 2x 5

Substituting the value of x:

Perimeter 2(5/√2) 5

This can be simplified to:

Perimeter 5/√2 5 5/√2

Combining like terms:

Perimeter 5 5√2

Converting to Decimal Form

To get a numerical value, we need to calculate the decimal form of √2.

√2 ≈ 1.414

So, the perimeter in decimal form is:

Perimeter ≈ 5 5 * 1.414 ≈ 5 7.07 ≈ 12.07 meters

Alternative Method

Another way to calculate the side length involves:

Given:

Hypotenuse 5m

Base/Height x √(25/2)

x ≈ 3.5355

Calculating the perimeter:

Base Height Hypotenuse ≈ 3.5355 3.5355 5 ≈ 12.071 meters

Conclusion

The perimeter of a right-angled isosceles triangle with a hypotenuse of 5 meters is approximately 12.071 meters. This detailed calculation demonstrates the application of the Pythagorean Theorem and provides a clear understanding of perimeter calculations for specific types of right-angled triangles.