An 8-inch by 8-inch square is folded along a diagonal creating a triangular region. This resulting triangular region is then folded so that the right angle vertex just meets the midpoint of the hypotenuse. What is the area of the resulting trapezoidal figure in square inches?
well, the diagonal has length 8√2
the altitude of the triangular region is thus half that, or 4√2
So, the trapezoid has bases 8√2 and 4√2 and altitude 2√2
The area is thus 6√2 * 2√2 = 24
Well, it sounds like we're doing some origami math here! Let me put on my paper folding hat.
First, we fold the 8-inch square along the diagonal, creating two congruent right triangles. Then, we fold one of those triangles so that the right angle vertex just meets the midpoint of the hypotenuse.
Now, since the right triangles are congruent, the leg of each triangle is 8/√2 inches long, which simplifies to approximately 5.66 inches.
When we fold the triangle, it forms a smaller right triangle with one leg measuring 5.66 inches and the other leg measuring half of the original leg, which is 2.83 inches.
Now, the area of the trapezoidal figure is the difference between the area of the original square and the area of the smaller right triangle.
The area of the original square is 8 inches multiplied by 8 inches, which gives us 64 square inches.
The area of the smaller right triangle is 1/2 times the base (2.83 inches) times the height (5.66 inches).
So, the area of the trapezoidal figure is 64 square inches minus (1/2) times 2.83 inches times 5.66 inches.
Area = 64 - (1/2)(2.83)(5.66)
Area = 64 - 8
Area = 56 square inches
So, the area of the resulting trapezoidal figure is 56 square inches. Happy origami math!
To find the area of the resulting trapezoidal figure, we need to find the length of the two bases and the height.
First, let's find the length of the two bases:
1. The length of the bottom base is equal to the length of the square, which is 8 inches.
2. To find the length of the top base, we need to find the length of the hypotenuse of the folded triangle. Since the square is folded along the diagonal, the hypotenuse of the triangle is equal to the length of the diagonal of the square.
Using the Pythagorean Theorem, we can calculate the length of the diagonal:
a² + b² = c² (where a and b are the sides of the square)
8² + 8² = c²
64 + 64 = c²
128 = c²
Taking the square root of both sides:
√128 = √c²
√128 = c
c ≈ 11.31 inches (rounded to two decimal places)
Therefore, the length of the top base of the trapezoidal figure is approximately 11.31 inches.
Now, let's find the height of the trapezoidal figure:
Since the right angle vertex of the folded triangle meets the midpoint of the hypotenuse, the height of the trapezoidal figure is half the length of the bottom base.
height = 8 inches / 2 = 4 inches
Now, we can calculate the area of the trapezoidal figure using the formula:
Area = (length of the top base + length of the bottom base) * height / 2
Area = (11.31 inches + 8 inches) * 4 inches / 2
Area = 19.31 inches * 4 inches / 2
Area = 77.24 square inches
Therefore, the area of the resulting trapezoidal figure is approximately 77.24 square inches.
To solve this problem, we can break it down into steps:
Step 1: Find the length of the diagonal of the square.
Since the square has sides of 8 inches, we can use the Pythagorean theorem to find the length of the diagonal. The diagonal is the hypotenuse of a right triangle with two sides measuring 8 inches. Using the Pythagorean theorem (a^2 + b^2 = c^2), we have:
8^2 + 8^2 = c^2
64 + 64 = c^2
128 = c^2
Taking the square root of both sides, we get:
c = √128
c ≈ 11.31 inches
Step 2: Fold the square along the diagonal to create a triangular region.
Fold the square along the diagonal so that two of the corners meet and form a right angle.
Step 3: Fold the triangular region so that the right angle vertex meets the midpoint of the hypotenuse.
Fold the triangular region so that the vertex of the right angle touches the midpoint of the hypotenuse. This will create a trapezoidal figure.
Step 4: Find the height of the trapezoidal figure.
The height of the trapezoidal figure is half of the length of the diagonal. Since the length of the diagonal is approximately 11.31 inches, the height of the trapezoidal figure is approximately 11.31/2 = 5.65 inches.
Step 5: Find the width of the trapezoidal figure.
The width of the trapezoidal figure is the length of one of the sides of the square. Since the sides of the square are 8 inches, the width of the trapezoidal figure is 8 inches.
Step 6: Find the area of the trapezoidal figure.
The area of a trapezoid can be calculated using the formula: area = (base1 + base2) * height / 2. In this case, the base1 and base2 are the width of the trapezoidal figure, which is 8 inches, and the height is 5.65 inches. Plugging in these values into the formula, we get:
area = (8 + 8) * 5.65 / 2
area = 16 * 5.65 / 2
area = 90.4 / 2
area ≈ 45.2 square inches
Therefore, the area of the resulting trapezoidal figure is approximately 45.2 square inches.