Question

two students calculated the area of a polygon in the coordinate plane using different strategies. explain the two methods. which method do you prefer and why? student 1 area of abcde area = 1 + 4+4 + 6+3 + 1.5+1 area = 20.5 square units student 2 area of abcde area = 36 - 4 - 1 - 1.5 - 2 - 3 - 4 area = 20.5 square units

Explanation:

Student 1's Method

Student 1 divided the polygon ABCDE into smaller non - overlapping geometric shapes (such as triangles and rectangles). Then, calculated the area of each of these smaller shapes separately. For example, if there is a triangle with base $b$ and height $h$, its area is $\frac{1}{2}bh$, and for a rectangle with length $l$ and width $w$, its area is $lw$. After finding the areas of all the smaller shapes, they summed them up: $1 + 4+4 + 6+3 + 1.5+1=20.5$ square units.

Student 2's Method

Student 2 enclosed the polygon ABCDE in a larger rectangle. Then, found the area of this larger rectangle. After that, calculated the areas of the non - polygon (empty) triangles and rectangles within the larger rectangle. Subtracted the sum of the areas of these non - polygon shapes from the area of the larger rectangle. The area of the larger rectangle is $6\times6 = 36$ square units. The sum of the areas of the non - polygon shapes is $4 + 1+1.5+2+3+4=15.5$ square units. So, the area of the polygon is $36-15.5 = 20.5$ square units.

Preference

The preference may vary. Student 1's method is good when the polygon can be easily decomposed into simple, well - known geometric shapes. It is a building - up approach. Student 2's method is useful when it is easy to enclose the polygon in a simple shape (like a rectangle) and the non - polygon parts are also simple geometric shapes. It is a subtraction approach. If the polygon has a complex shape but can be easily enclosed in a regular shape, Student 2's method might be preferred as it can sometimes reduce the number of individual area calculations. However, if the polygon has natural break - points for division into simple shapes, Student 1's method is more straightforward.

Answer:

Student 1: Decomposed polygon into smaller shapes and summed their areas. Student 2: Enclosed polygon in a rectangle and subtracted the areas of non - polygon parts. Preference depends on polygon's shape; if easily decomposable, prefer Student 1's method, if easily enclosable, prefer Student 2's method.