Question

consider the polygon below.
image of a polygon with labeled sides: x-3 (top left vertical), x-3 (top right vertical), 2x-3 (top middle horizontal), x-2 (middle vertical), x+3 (right vertical), and bottom horizontal (length to be determined?)
determine the total area of this polygon as a simplified algebraic expression.
○ 2x² - 4x - 21
○ 2x² - 4x - 33
○ 2x² - 4x - 33 (wait, no, the options are: 2x² - 4x - 21, 2x² - 4x - 33, 6x² - 4x - 21, 2x² + 10x - 33)

Explanation:

Step1: Find the length of the base

The base length is the sum of the two \(x - 3\) segments and the middle \(2x - 3\) segment. So, base \(= (x - 3)+(2x - 3)+(x - 3)=x - 3+2x - 3+x - 3 = 4x - 9\)? Wait, no, maybe a better approach: the polygon can be seen as a large rectangle minus a smaller rectangle. Let's find the dimensions of the large rectangle. The height of the large rectangle is \(x + 3\). The length of the large rectangle: let's see, the top has two segments of \(x - 3\) and one of \(2x - 3\), so total length \(L=(x - 3)+(2x - 3)+(x - 3)=4x - 9\)? Wait, no, maybe I made a mistake. Alternatively, the large rectangle (if we fill the indentation) would have length \( (x - 3)+(2x - 3)+(x - 3)=4x - 9\)? No, wait, maybe the large rectangle's length is \( (x - 3)+(2x - 3)+(x - 3)=x - 3+2x - 3+x - 3 = 4x - 9\), and height \(x + 3\). Then the area of the large rectangle is \(L\times H=(4x - 9)(x + 3)\). Now, the area of the indentation (the smaller rectangle we need to subtract) has length \(2x - 3\) and height \(x - 2\). So area of indentation is \((2x - 3)(x - 2)\). Then total area \(A=(4x - 9)(x + 3)-(2x - 3)(x - 2)\). Wait, maybe another approach: split the polygon into three rectangles. The left rectangle: length \(x - 3\), height \(x + 3\). The middle indentation? No, wait, the polygon has two vertical rectangles on the left and right, and a horizontal rectangle at the bottom? Wait, looking at the diagram: the left part is a rectangle with width \(x - 3\) and height \(x + 3\). The right part is the same: width \(x - 3\), height \(x + 3\). The bottom middle part: width \(2x - 3\), height \((x + 3)-(x - 2)=x + 3 - x + 2 = 5\)? Wait, no, the height of the bottom middle rectangle: the total height is \(x + 3\), and the height of the top indentation is \(x - 2\), so the bottom middle rectangle has height \(x + 3-(x - 2)=5\). So area of left rectangle: \((x - 3)(x + 3)=x^2 - 9\) (using difference of squares). Area of right rectangle: same as left, \(x^2 - 9\). Area of bottom middle rectangle: \((2x - 3)\times5 = 10x - 15\). Wait, but that doesn't seem right. Wait, maybe the correct way is to consider the polygon as a large rectangle minus a smaller rectangle. Let's find the length of the large rectangle: the top has two segments of \(x - 3\) and one of \(2x - 3\), so total length \(L=(x - 3)+(2x - 3)+(x - 3)=4x - 9\)? No, wait, maybe the large rectangle's length is \( (x - 3)+(2x - 3)+(x - 3)=4x - 9\), and height \(x + 3\). Then area of large rectangle: \((4x - 9)(x + 3)=4x^2 + 12x - 9x - 27=4x^2 + 3x - 27\). Then the area of the indentation (the part we need to subtract) is a rectangle with length \(2x - 3\) and height \(x - 2\), so area is \((2x - 3)(x - 2)=2x^2 - 4x - 3x + 6=2x^2 - 7x + 6\). Then total area \(A=4x^2 + 3x - 27-(2x^2 - 7x + 6)=4x^2 + 3x - 27 - 2x^2 + 7x - 6=2x^2 + 10x - 33\). Wait, but let's check again. Wait, maybe my initial approach was wrong. Let's split the polygon into three rectangles: left, middle bottom, right. Left rectangle: width \(x - 3\), height \(x + 3\). Area: \((x - 3)(x + 3)=x^2 - 9\). Right rectangle: same as left, \(x^2 - 9\). Middle bottom rectangle: width \(2x - 3\), height \((x + 3)-(x - 2)=5\) (since the top indentation has height \(x - 2\), so the bottom middle has height total height minus indentation height). So area of middle bottom: \((2x - 3)\times5=10x - 15\). Now, total area: \(x^2 - 9 + x^2 - 9 + 10x - 15=2x^2 + 10x - 33\). Yes, that matches. Let's verify with the other method. Large rectangle: length is \((x - 3)+(2x - 3)+(x - 3)=4x - 9\), height \(x + 3\). Area: \((4x - 9)(x + 3)=4x^2 + 12x - 9x - 27=4x^2 + 3x - 27\). Indentation area: length \(2x - 3\), height \(x - 2\). Area: \((2x - 3)(x - 2)=2x^2 - 4x - 3x + 6=2x^2 - 7x + 6\). Subtract: \(4x^2 + 3x - 27 - (2x^2 - 7x + 6)=4x^2 + 3x - 27 - 2x^2 + 7x - 6=2x^2 + 10x - 33\). So the total area is \(2x^2 + 10x - 33\).

Answer:

\(2x^2 + 10x - 33\) (corresponding to the option: \(2x^2 + 10x - 33\))