Question

graph the following figure in the coordinate plane. find the perimeter. x(0,1), y(4, - 2), z(-5, - 2) choose the correct graph below.

Explanation:

Step1: Recall the distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate the distance between $X(0,1)$ and $Y(4, - 2)$

$d_{XY}=\sqrt{(4 - 0)^2+(-2 - 1)^2}=\sqrt{4^2+( - 3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$.

Step3: Calculate the distance between $Y(4,-2)$ and $Z(-5,-2)$

$d_{YZ}=\sqrt{(-5 - 4)^2+(-2+2)^2}=\sqrt{(-9)^2+0^2}=\sqrt{81}=9$.

Step4: Calculate the distance between $Z(-5,-2)$ and $X(0,1)$

$d_{ZX}=\sqrt{(0 + 5)^2+(1 + 2)^2}=\sqrt{5^2+3^2}=\sqrt{25 + 9}=\sqrt{34}$.

Step5: Calculate the perimeter

The perimeter $P=d_{XY}+d_{YZ}+d_{ZX}=5 + 9+\sqrt{34}=14+\sqrt{34}$.

To graph the points:

• The point $X(0,1)$ is on the $y$ - axis, 1 unit above the origin.
• The point $Y(4,-2)$ is 4 units to the right and 2 units down from the origin.
• The point $Z(-5,-2)$ is 5 units to the left and 2 units down from the origin.

We need to match the graph with the correct placement of these points. Without seeing all the options in detail, we can analyze the relative positions of the points. The points $Y(4,-2)$ and $Z(-5,-2)$ have the same $y$ - coordinate, so they lie on a horizontal line.

Answer:

We first find the perimeter as $14+\sqrt{34}$. To choose the correct graph, look for a graph where the point $X$ is on the $y$ - axis at $y = 1$, point $Y$ is 4 units to the right and 2 units down from the origin, and point $Z$ is 5 units to the left and 2 units down from the origin with $Y$ and $Z$ on a horizontal line.