Victor cut four congruent triangles off the corners of a rectangle to make a hexagon, as shown below.
8 cm
A
16 cm
8 cm
28 cm
What is the area of the shaded hexagon?
O A. 320 cm²
OB. 384 cm²
О C. 416 cm²
D.
576 cm²
O
C. 416 cm²
sorry, STARZ -- too many problems too fast
nice try.
To find the area of the shaded hexagon, we need to find the area of the rectangle and subtract the area of the four congruent triangles.
Step 1: Find the area of the rectangle.
The rectangle has dimensions 16 cm by 28 cm, so the area is A = length x width = 16 cm x 28 cm = 448 cm².
Step 2: Find the area of one triangle.
Since the triangles are congruent, we only need to find the area of one triangle.
The base of the triangle is 8 cm, and the height can be calculated using the Pythagorean theorem.
The height is the square root of the square of half the base (which is 4 cm) plus the square of the longest side (which is 28 cm).
Height = √(4² + 28²) = √(16 + 784) = √800 = 20√2 cm.
The area of one triangle is A = (1/2) x base x height = (1/2) x 8 cm x 20√2 cm = 80√2 cm².
Step 3: Find the area of all four triangles.
Since there are four triangles, the total area of the triangles is 4 times the area of one triangle.
Total triangle area = 4 x 80√2 cm² = 320√2 cm².
Step 4: Find the area of the shaded hexagon.
To find the area of the shaded hexagon, subtract the total triangle area from the area of the rectangle.
Shaded hexagon area = Rectangle area - Total triangle area = 448 cm² - 320√2 cm².
Now, we need to determine whether there is enough information to simplify the expression for the shaded hexagon area.
Unfortunately, we do not have the value for √2, so we cannot simplify the expression further. Therefore, the area of the shaded hexagon is 448 cm² - 320√2 cm².
Therefore, the correct answer is D. 448 cm² - 320√2 cm².
To find the area of the shaded hexagon, we can divide it into smaller triangles and rectangles.
First, let's calculate the area of the rectangle. The dimensions of the rectangle are given as 16 cm by 28 cm, so its area is:
Area of rectangle = length × width = 16 cm × 28 cm = 448 cm²
Now, let's calculate the area of each of the four congruent triangles that were cut off the corners of the rectangle. The dimensions of each triangle are:
Base = 8 cm
Height = 8 cm
The area of a triangle is given by the formula:
Area of triangle = 1/2 × base × height
So, the area of each triangle is:
Area of triangle = 1/2 × 8 cm × 8 cm = 32 cm²
Since there are four congruent triangles, the total area of all four triangles is:
Total area of triangles = 4 × 32 cm² = 128 cm²
Now, we can find the area of the shaded hexagon by subtracting the area of the four triangles from the area of the rectangle:
Area of shaded hexagon = Area of rectangle - Total area of triangles
= 448 cm² - 128 cm²
= 320 cm²
Therefore, the area of the shaded hexagon is 320 cm².
So, the correct option is A. 320 cm².