a garden is shaped like an x. there is a right angle at each corner of the garden and each side of the garden measures 40 feet long. what is the area of this garden?
To find the area of the garden shaped like an X, we need to determine the area of each individual section and then add them together. Since the garden is shaped like an X and has four right angles, it consists of four congruent right-angled triangles.
To calculate the area of a triangle, we need the length of its base and height. In this case, the base is one of the sides of the garden, which measures 40 feet long. The height of the triangle is the distance from the center of the X to one of the corners, and since the garden is symmetrical, all four triangles have the same height.
To determine the height, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Since we know the length of the base (40 feet) and the length of the hypotenuse (also 40 feet, as it is the side of the garden), we can solve for the height.
Let's denote the height of the triangle as 'h'. Using the Pythagorean theorem:
h^2 + (40/2)^2 = 40^2
Simplifying:
h^2 + 20^2 = 40^2
h^2 + 400 = 1600
h^2 = 1600 - 400
h^2 = 1200
h = √1200
h ≈ 34.64 feet
Now that we have the height, we can calculate the area of each triangle:
Area of a triangle = (base * height) / 2
Area of each triangle = (40 * 34.64) / 2 ≈ 692.8 square feet
Since there are four identical triangles, we multiply the area of each triangle by 4 to find the total area of the garden:
Total area of the garden = 4 * 692.8 ≈ 2771.2 square feet
Therefore, the area of the garden shaped like an X is approximately 2771.2 square feet.