Question

question 7 of 10
what is the area of the polygon given below?
5
6
12
10
6
a. 90 square units
b. 60 square units
c. 120 square units
d. 180 square units

Explanation:

Step1: Divide the polygon

We can divide the L - shaped polygon into two rectangles. One rectangle has dimensions \(5\times12\) and the other has dimensions \(10\times6\). But wait, actually, a better way is to see that the height of the first rectangle (left - most) is \(12\) and width \(5\), and the second rectangle (right - most) has length \(10\) and width \(6\), but also, we can think of the total length in the horizontal direction as \(5 + 10=15\) and total height in vertical direction? No, better to split into two rectangles: one with length \(5\) and height \(12\), and the other with length \(10\) and height \(6\). Wait, no, actually, the vertical side of the first rectangle is \(12\), and the horizontal side is \(5\). The second rectangle: the horizontal side is \(10\) and the vertical side is \(6\) (since the total height is \(12\), and the upper part of the second rectangle is \(12 - 6 = 6\)? Wait, no, let's look at the figure again. The polygon has a vertical side of \(12\), a horizontal top side of \(5\), then a horizontal segment of \(10\), and a vertical segment of \(6\) down, and then a horizontal segment back? Wait, the correct way is to split the L - shape into two rectangles. One rectangle has length \(5\) and height \(12\), and the other rectangle has length \(10\) and height \(6\). Wait, no, actually, the height of the second rectangle (the one with length \(10\)) is \(6\), and the first rectangle (with length \(5\)) has height \(12\). But another way: the total area can be calculated as the area of a rectangle with length \(5 + 10=15\) and height \(6\) plus the area of a rectangle with length \(5\) and height \(12 - 6 = 6\). Let's calculate both ways.

First way: Split into two rectangles. Rectangle 1: \(l_1 = 5\), \(h_1=12\), area \(A_1=5\times12 = 60\). Rectangle 2: \(l_2 = 10\), \(h_2 = 6\), area \(A_2=10\times6=60\). Then total area \(A = A_1+A_2=5\times12+10\times6=60 + 60=120\)? Wait, no, wait, that's wrong. Wait, actually, the correct split is: the first rectangle is \(5\times(12 - 6)=5\times6 = 30\) and the second rectangle is \((5 + 10)\times6=15\times6 = 90\). Wait, no, let's do it properly.

Looking at the figure, the vertical side is \(12\), horizontal side at the top is \(5\), then a horizontal line of \(10\) to the right, then a vertical line down of \(6\), then a horizontal line to the left? Wait, no, the figure has: from the bottom left, go up \(12\), right \(5\), down \(6\), right \(10\), down \(6\), left \(15\) (since \(5 + 10 = 15\))? No, that can't be. Wait, the correct dimensions: the total height is \(12\), the width of the left rectangle is \(5\), and the width of the right rectangle is \(10\), and the height of the right rectangle is \(6\), and the height of the left rectangle is \(12\). Wait, no, the right rectangle's height is \(6\), and the left rectangle's height is \(12\), and the left rectangle's width is \(5\), the right rectangle's width is \(10\). But actually, the area of the left rectangle is \(5\times12 = 60\), and the area of the right rectangle is \(10\times6=60\), so total area \(60 + 60=120\)? Wait, no, that's not right. Wait, another approach: the big rectangle that would enclose the L - shape has length \(5 + 10 = 15\) and height \(12\), but then we subtract the area of the missing part? No, the missing part is a rectangle with length \(10\) and height \(12 - 6=6\). Wait, the area of the big rectangle (if it was a full rectangle) is \(15\times12=180\), and the missing area is \(10\times6 = 60\), so \(180-60 = 120\). Wait, no, that's not correct. Wait, let's look at the figure again. The vertical side is \(12\), horizontal top segment \(5\), then a horizontal segment \(10\) to the right, then a vertical segment \(6\) down, then a horizontal segment to the left (length \(5 + 10 = 15\)) and a vertical segment \(6\) up? No, the correct way is to split the L - shape into two rectangles:

Rectangle 1: width \(5\), height \(12\). Area \(A_1=5\times12 = 60\).

Rectangle 2: width \(10\), height \(6\). Area \(A_2=10\times6 = 60\).

Wait, no, because the height of the second rectangle (the one with width \(10\)) is \(6\), and the first rectangle (width \(5\)) has height \(12\). But actually, the total area is \(A_1+A_2=60 + 60=120\). Wait, let's check with another split. The horizontal length of the bottom side is \(5 + 10=15\), and the height of the bottom rectangle is \(6\), and the height of the top rectangle is \(12 - 6 = 6\), and width \(5\). So area of bottom rectangle: \(15\times6=90\), area of top rectangle: \(5\times6 = 30\), total area \(90 + 30=120\). Yes, that's correct.

Wait, let's calculate using the first split:

Wait, maybe I made a mistake earlier. Let's define the two rectangles properly. The left rectangle has dimensions \(5\) (width) and \(12\) (height). The right rectangle has dimensions \(10\) (width) and \(6\) (height). But wait, the height of the right rectangle is \(6\), and the height of the left rectangle is \(12\), and the width of the left rectangle is \(5\), the width of the right rectangle is \(10\). But when we put them together, the total width is \(5 + 10 = 15\) and the total height is \(12\), but the right rectangle is attached to the left rectangle. Wait, no, the right rectangle is below the left rectangle? No, the figure is an L - shape, so the left part is a vertical rectangle of \(5\times12\), and the right part is a horizontal rectangle of \(10\times6\) attached to the bottom of the left rectangle? No, the vertical side is \(12\), so the left rectangle is \(5\) (width) by \(12\) (height), and the right rectangle is \(10\) (width) by \(6\) (height), and the bottom of the right rectangle is at the same level as the bottom of the left rectangle. So the total height of the left rectangle is \(12\), and the right rectangle has height \(6\), but the left rectangle's bottom is at the same level as the right rectangle's bottom. Wait, no, the vertical segment on the right is \(6\), so the height of the right rectangle is \(6\), and the height of the left rectangle is \(12\), and the horizontal distance between them is \(10\). So the area of the left rectangle is \(5\times12 = 60\), and the area of the right rectangle is \(10\times6=60\), so total area \(60 + 60 = 120\).

Step2: Calculate the area

Using the two - rectangle method:

First rectangle: length \(l_1 = 5\), height \(h_1=12\), area \(A_1=l_1\times h_1=5\times12 = 60\).

Second rectangle: length \(l_2 = 10\), height \(h_2=6\), area \(A_2=l_2\times h_2=10\times6 = 60\).

Total area \(A=A_1 + A_2=60+60 = 120\) square units.

Answer:

C. 120 square units