Question

question 21 of 25
what is the area of the polygon below?
image of an l - shaped polygon with dimensions: vertical side 10, bottom side 12, horizontal segment 9, vertical segment 6
a. 174 square units
b. 66 square units
c. 120 square units
d. 84 square units

Explanation:

Step1: Split the polygon

We can split the L - shaped polygon into two rectangles. Let's consider the vertical part and the horizontal part. The total length of the base is 12, and the horizontal segment on the top right is 9, so the length of the vertical rectangle's base is \(12 - 9=3\). The height of the vertical rectangle is 10, and the height of the horizontal rectangle is 6.

Step2: Calculate area of each rectangle

• Area of the vertical rectangle: The dimensions are length \(l_1 = 3\) and height \(h_1=10\). The area formula for a rectangle is \(A = l\times h\), so \(A_1=3\times10 = 30\).
• Area of the horizontal rectangle: The dimensions are length \(l_2 = 12\) and height \(h_2 = 6\)? Wait, no, that's wrong. Wait, the horizontal rectangle has length 9 and height 6? Wait, no, let's re - examine. The correct way: The polygon can be split into a rectangle with length 12 and height 6, and another rectangle with length \(10 - 6 = 4\) and width \(12 - 9=3\)? No, maybe a better way. Let's take the first rectangle as having length 9 and height 6, and the second rectangle as having length 12 and height \(10 - 6 = 4\). Wait, no, let's do it properly.

Alternative split: One rectangle is \(12\times6\) and the other is \(3\times(10 - 6)\).
Area of the first rectangle ( \(12\times6\)): \(A_1=12\times6 = 72\).
Area of the second rectangle: The width is \(12 - 9 = 3\) and the height is \(10 - 6=4\)? No, wait, the vertical side is 10, and the lower rectangle has height 6, so the upper rectangle has height \(10 - 6 = 4\) and width 3 (since \(12-9 = 3\)). So area of the second rectangle \(A_2=3\times4=12\)? No, that's not right. Wait, another way: The polygon can be considered as a large rectangle of \(12\times10\) minus a smaller rectangle of \(9\times(10 - 6)\).
Large rectangle area: \(A_{large}=12\times10=120\).
Smaller rectangle: The length is 9 and the height is \(10 - 6 = 4\), so area \(A_{small}=9\times4 = 36\). Then the area of the polygon is \(A = 120-36=84\). Or, split into two rectangles: One with length 12 and height 6 (\(A_1 = 12\times6 = 72\)) and another with length \(12 - 9=3\) and height \(10 - 6 = 4\)? No, \(3\times4 = 12\), and \(72 + 12=84\). Or, the other rectangle is length 9 and height 6 (\(A_1=9\times6 = 54\)) and the other rectangle is length 12 and height \(10 - 6 = 4\) (\(A_2=12\times4 = 48\))? No, that would be wrong. Wait, no, the correct split is: The polygon is composed of a rectangle with length 9 and height 6, and a rectangle with length 12 and height \(10 - 6 = 4\)? No, \(9\times6=54\), \(12\times4 = 48\), \(54 + 48=102\), which is not one of the options. I must have made a mistake.
Wait, let's look at the options. The options are 174, 66, 120, 84. Let's try another approach. The total height is 10, total length is 12. If we consider the polygon as a rectangle of \(12\times10\) minus a rectangle of \(9\times(10 - 6)\). \(12\times10 = 120\), \(10 - 6 = 4\), \(9\times4 = 36\), \(120-36 = 84\). Yes, that works.

Step3: Verify the area

So the area of the polygon is the area of the large rectangle (\(12\times10\)) minus the area of the missing rectangle (\(9\times(10 - 6)\)).

• Area of large rectangle: \(A_{total}=12\times10 = 120\)
• Area of missing rectangle: The height of the missing rectangle is \(10 - 6=4\) (since the lower part has height 6), and the length is 9. So \(A_{missing}=9\times4 = 36\)
• Area of the polygon: \(A = 120-36=84\) square units.

Answer:

D. 84 square units