Complete Question: You are building a shelf that fits in a corner in the figure the entire shelf is XYZ each unit in the coordinate plane represents one inch find the area of the shelf.
Answer:
112.5 units²
Step-by-step explanation:
Area of the shelf = ½*base*height.
The base is the distance between Z(0, 5) and Y(15, 5)
The height is the distance between X(15, 20) and Y(15, 5).
Therefore area of the shelf = ½*ZY*XY
Use the distance formula to find ZY and XY.
Distance between Z(0, 5) and Y(15, 5):
![ZY = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}](https://tex.z-dn.net/?f=%20ZY%20%3D%20%5Csqrt%7B%28x_2%20-%20x_1%29%5E2%20%2B%20%28y_2%20-%20y_1%29%5E2%7D%20)
Let,
![Z(0, 5) = (x_1, y_1)](https://tex.z-dn.net/?f=%20Z%280%2C%205%29%20%3D%20%28x_1%2C%20y_1%29%20)
![Y(15, 5) = (x_2, y_2)](https://tex.z-dn.net/?f=%20Y%2815%2C%205%29%20%3D%20%28x_2%2C%20y_2%29%20)
![ZY = \sqrt{(15 - 0)^2 + (5 - 5)^2}](https://tex.z-dn.net/?f=%20ZY%20%3D%20%5Csqrt%7B%2815%20-%200%29%5E2%20%2B%20%285%20-%205%29%5E2%7D%20)
![AB = \sqrt{(15)^2 + (0)^2}](https://tex.z-dn.net/?f=%20AB%20%3D%20%5Csqrt%7B%2815%29%5E2%20%2B%20%280%29%5E2%7D%20)
![AB = \sqrt{225 + 0} = \sqrt{225}](https://tex.z-dn.net/?f=%20AB%20%3D%20%5Csqrt%7B225%20%2B%200%7D%20%3D%20%5Csqrt%7B225%7D%20)
![AB = 15](https://tex.z-dn.net/?f=%20AB%20%3D%2015%20)
Distance between X(15, 20) and Y(15, 5):
![XY = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}](https://tex.z-dn.net/?f=%20XY%20%3D%20%5Csqrt%7B%28x_2%20-%20x_1%29%5E2%20%2B%20%28y_2%20-%20y_1%29%5E2%7D%20)
Let,
![X(15, 20) = (x_1, y_1)](https://tex.z-dn.net/?f=%20X%2815%2C%2020%29%20%3D%20%28x_1%2C%20y_1%29%20)
![Y(15, 5) = (x_2, y_2)](https://tex.z-dn.net/?f=%20Y%2815%2C%205%29%20%3D%20%28x_2%2C%20y_2%29%20)
![XY = \sqrt{(15 - 15)^2 + (5 - 20)^2}](https://tex.z-dn.net/?f=%20XY%20%3D%20%5Csqrt%7B%2815%20-%2015%29%5E2%20%2B%20%285%20-%2020%29%5E2%7D%20)
![XY = \sqrt{(0)^2 + (-15)^2}](https://tex.z-dn.net/?f=%20XY%20%3D%20%5Csqrt%7B%280%29%5E2%20%2B%20%28-15%29%5E2%7D%20)
![XY = \sqrt{0 + 225} = \sqrt{225}](https://tex.z-dn.net/?f=%20XY%20%3D%20%5Csqrt%7B0%20%2B%20225%7D%20%3D%20%5Csqrt%7B225%7D%20)
![XY = 15](https://tex.z-dn.net/?f=%20XY%20%3D%2015%20)
Area of shelf = ½*15*15 = ½*225 = 112.5 units²