To find the perimeter of an isosceles right angled triangle given its area, we start with the known properties of such triangles.
The area (A) of an isosceles right angled triangle can be calculated using the formula:
A = (1/2) * base * height
In an isosceles right angled triangle, the base and height are equal, so we can express the area as:
A = (1/2) * a * a = (1/2) * a²
Where a is the length of each equal side of the triangle.
Given that the area is 5000 m², we can set up the equation:
5000 = (1/2) * a²
Multiplying both sides by 2 gives us:
10000 = a²
Taking the square root of both sides, we find:
a = √10000 = 100 m
Now that we have the length of the equal sides of the triangle, we can calculate the perimeter (P). The perimeter of an isosceles right angled triangle is given by:
P = a + a + (a√2) = 2a + a√2
Substituting the value of a:
P = 2(100) + 100√2
This simplifies to:
P = 200 + 100√2
Calculating the numerical value (using √2 ≈ 1.414):
P ≈ 200 + 100(1.414) ≈ 200 + 141.4 ≈ 341.4 m
Thus, the perimeter of the isosceles right angled triangle is approximately 341.4 meters.