Given that the side of the square has a length b-a, find the area of one of the four triangles and the area of the small inner square.
Give the area of one of the triangles followed by the area of the small inner square separated by a comma. Express your answers in terms of the variables a and b.
What triangles? What small inner square?
To find the area of one of the four triangles, we need to determine the length of its base and height.
The base of the triangle is the same as the side length of the square, which is b-a. Since the triangle is located on one side of the square, its height is equal to the side length of the square, b-a.
Therefore, the area of one of the triangles can be calculated using the formula for the area of a triangle:
Area of a triangle = (1/2) x base x height
Area of a triangle = (1/2) x (b-a) x (b-a)
Area of a triangle = (1/2)(b - a)^2
On the other hand, the area of the small inner square can be found by calculating the square of the side length, which is also b-a.
Area of the small inner square = (b-a)^2
So the area of one of the triangles is equal to (1/2)(b - a)^2, and the area of the small inner square is (b-a)^2.
Therefore, the area of one of the triangles followed by the area of the small inner square is:
(1/2)(b - a)^2, (b-a)^2
To find the area of one of the triangles, we need to know the length of its base and height. The base of the triangle is the side of the square, so it has a length of b-a.
Now, let's find the height of the triangle. The height is the distance from the base to the opposite vertex of the square. Since the square has four equal sides, the height can be determined by dividing the side length, b-a, by the square root of 2 (√2).
Therefore, the height of the triangle is (b-a)/√2.
To calculate the area of the triangle, we use the formula A = (1/2) * base * height, where A is the area, base is the length of the base, and height is the height of the triangle.
Putting it all together, the area of one of the triangles is:
A_triangle = (1/2) * (b-a) * (b-a)/√2, which simplifies to:
A_triangle = (b-a)^2 / (2√2).
Next, let's find the area of the small inner square. The side length of the small square is equal to the height of the triangle, which is (b-a)/√2.
To calculate the area of a square, we use the formula A = side^2.
Therefore, the area of the small inner square is:
A_square = [(b-a)/√2]^2, which simplifies to:
A_square = (b-a)^2 / 2.
So, the area of one of the triangles is (b-a)^2 / (2√2), and the area of the small inner square is (b-a)^2 / 2.
Final answer:
Area of one triangle, A_triangle = (b-a)^2 / (2√2).
Area of the small inner square, A_square = (b-a)^2 / 2.