Question

1. windows the shape of a boat window is a parallelogram. the four vertices of the parallelogram on the coordinate plane are shown in the figure. each unit on the coordinate plane represents 1 inch. what is the perimeter of the window? what is the area? round your answers to the nearest tenth, if necessary.

Explanation:

Step1: Identify the base and height of the parallelogram

From the graph, the base \( b \) of the parallelogram (the horizontal side) is 20 inches (from \( x = 0 \) to \( x = 20 \) for the bottom side? Wait, no, looking at the coordinates: the bottom vertices seem to be at \( (0,0) \) and \( (20,0) \), and the top vertices at \( (2,18) \) and \( (22,18) \)? Wait, maybe better to find the length of the sides. The base length: the distance between \( (0,0) \) and \( (20,0) \) is \( 20 - 0 = 20 \) inches. Now, the height: the vertical distance from the base to the top side. The y - coordinate of the top side is 18, and the base is at y = 0, so height \( h = 18 \) inches? Wait, no, maybe the slant side. Wait, let's find the length of the slant side. Take two points: \( (0,0) \) and \( (2,18) \). The distance between \( (x_1,y_1)=(0,0) \) and \( (x_2,y_2)=(2,18) \) is \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{(2 - 0)^2+(18 - 0)^2}=\sqrt{4 + 324}=\sqrt{328}\approx18.11 \) inches.

Step2: Calculate the perimeter of the parallelogram

The perimeter \( P \) of a parallelogram is given by \( P = 2\times(\text{base}+\text{slant side}) \). We know the base \( b = 20 \) inches and the slant side \( s\approx18.11 \) inches. So \( P = 2\times(20 + 18.11)=2\times(38.11) = 76.22\approx76.2 \) inches? Wait, wait, maybe I misread the coordinates. Wait, looking at the graph, the top side is from \( x = 2 \) to \( x = 22 \)? Wait, the right - bottom vertex is at \( (20,0) \), and the right - top vertex is at \( (22,18) \)? Wait, no, the graph shows the top side from \( x = 2 \) to \( x = 22 \) (since the y - axis is up to 20, and the top points are at y = 18). Wait, the base: the bottom side is from \( (0,0) \) to \( (20,0) \), length 20. The top side is from \( (2,18) \) to \( (22,18) \), length \( 22 - 2 = 20 \), so that's consistent with a parallelogram (opposite sides equal). Now, the left side: from \( (0,0) \) to \( (2,18) \), and the right side: from \( (20,0) \) to \( (22,18) \). Let's recalculate the length of the left side. The horizontal distance between \( (0,0) \) and \( (2,18) \) is \( 2 - 0 = 2 \), vertical distance is \( 18 - 0 = 18 \). So length \( l=\sqrt{2^{2}+18^{2}}=\sqrt{4 + 324}=\sqrt{328}\approx18.11 \).

So perimeter \( P = 2\times(20 + 18.11)=2\times38.11 = 76.22\approx76.2 \) inches.

Step3: Calculate the area of the parallelogram

The area \( A \) of a parallelogram is given by \( A=\text{base}\times\text{height} \). The base is 20 inches (horizontal length), and the height is the vertical distance between the two horizontal sides. The vertical distance is the difference in y - coordinates, which is \( 18 - 0 = 18 \) inches (since the base is on \( y = 0 \) and the top side is on \( y = 18 \)). So \( A=20\times18 = 360 \) square inches.

Answer:

Perimeter: \(\approx76.2\) inches, Area: \(360\) square inches