Question

Cedric drew four polygons and determined their measurements as shown below. Use decomposition to find the area of each polygon, and then place the polygons in order from greatest to least area.

1. Rhombus: Slanting height = 5 inches, Diagonal height = 4 inches, Base = 3 inches
2. Parallelogram: Length = 7 inches, Slanting height = 3.6 inches, Diagonal height = 3 inches, Edge to diagonal line = 2 inches
3. Isosceles trapezoid: Base = 9 inches, Slanting height = 6.4 inches, Height = 5 inches, Length at the top = 1 inch
4. Polygon: Top = 6 inches, Extended width on the left, Width = 5 inches, Width on the right = 3 inches, Length at the bottom = 4 inches

Arrange the polygons in order from greatest to least area.

Answer 1

To find the area of each polygon, we need to first decompose them into simpler shapes. The areas of the given polygons are as follows: Rhombus - 7.5 square inches, Parallelogram - 25.2 square inches, Isosceles Trapezoid - 25 square inches, Polygon - 42 square inches. Arranging them in order from greatest to least area, we get: Polygon, Isosceles Trapezoid, Parallelogram, Rhombus.

Step-by-step explanation:

To find the area of each polygon, we need to first decompose them into simpler shapes. Let's start with the given information:

1. Rhombus: Slanting height = 5 inches, Diagonal height = 4 inches, Base = 3 inches
2. Parallelogram: Length = 7 inches, Slanting height = 3.6 inches, Diagonal height = 3 inches, Edge to diagonal line = 2 inches
3. Isosceles trapezoid: Base = 9 inches, Slanting height = 6.4 inches, Height = 5 inches, Length at the top = 1 inch
4. Polygon: Top = 6 inches, Extended width on the left, Width = 5 inches, Width on the right = 3 inches, Length at the bottom = 4 inches

Now, let's calculate the area for each polygon:

1. The area of a rhombus can be found using the formula A = (base x slanting height) / 2. So, for the given rhombus, the area is (3 inches x 5 inches) / 2 = 7.5 square inches.
2. The area of a parallelogram can be found using the formula A = base x slanting height. So, the area of the given parallelogram is 7 inches x 3.6 inches = 25.2 square inches.
3. The area of an isosceles trapezoid can be found using the formula A = ((length of top + length of bottom)/2) x height. So, the area of the given trapezoid is ((1 inch + 9 inches)/2) x 5 inches = 25 square inches.
4. The area of the given polygon can be found by dividing it into smaller shapes. From the given information, it seems that the polygon can be divided into a rectangle and two right triangles. The area of the rectangle is length x width, which in this case is 4 inches x 6 inches = 24 square inches. The area of each right triangle can be found using the formula A = (base x height)/2. The base and height of each triangle are 3 inches and 6 inches, so the area of each triangle is (3 inches x 6 inches)/2 = 9 square inches. Adding the areas of the rectangle and two triangles, we get a total area of 24 square inches + 9 square inches + 9 square inches = 42 square inches for the given polygon.

Now that we have found the area for each polygon, we can arrange them in order from greatest to least area:

1. Polygon - 42 square inches
2. Isosceles Trapezoid - 25 square inches
3. Parallelogram - 25.2 square inches
4. Rhombus - 7.5 square inches