Questions: x=(0,1) y=(4,-2) z=(-5,-2) The perimeter of xyz is how many units?

Solution Steps

To find the perimeter of the triangle formed by the points \( x \), \( y \), and \( z \), we need to calculate the distances between each pair of points and then sum these distances. The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the distance formula: \[ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Using the distance formula: \[ d_{xy} = \sqrt{(4 - 0)^2 + (-2 - 1)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5.0 \]

Using the distance formula: \[ d_{yz} = \sqrt{(-5 - 4)^2 + (-2 - (-2))^2} = \sqrt{(-9)^2 + 0^2} = \sqrt{81} = 9.0 \]

Using the distance formula: \[ d_{zx} = \sqrt{(0 - (-5))^2 + (1 - (-2))^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.831 \]

Sum the distances calculated: \[ \text{Perimeter} = d_{xy} + d_{yz} + d_{zx} = 5.0 + 9.0 + 5.831 \approx 19.831 \]

Final Answer