Question

find the perimeter and area of the figure if each unit on the graph measures 1 centimeter. round answers to the nearest tenth, if necessary. perimeter: cm area: cm²

Explanation:

Step1: Calculate the length of one side using distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $G(3,-1)$ and $F(5,-3)$, $x_1 = 3,y_1=-1,x_2 = 5,y_2=-3$. Then $d=\sqrt{(5 - 3)^2+(-3+1)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$. Since the figure is a rhombus (by symmetry), all sides are equal.

Step2: Calculate the perimeter

The perimeter $P$ of a rhombus with side - length $s$ is $P = 4s$. Here $s = 2\sqrt{2}$, so $P=4\times2\sqrt{2}=8\sqrt{2}\approx11.3$ cm.

Step3: Calculate the area

The area $A$ of a rhombus is given by $A=\frac{1}{2}d_1d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals. The length of diagonal $HF$: Using the distance formula for $H(1,-3)$ and $F(5,-3)$, $d_{HF}=\sqrt{(5 - 1)^2+(-3 + 3)^2}=4$. The length of diagonal $GE$: Using the distance formula for $G(3,-1)$ and $E(3,-5)$, $d_{GE}=\sqrt{(3 - 3)^2+(-5 + 1)^2}=4$. Then $A=\frac{1}{2}\times4\times4 = 8$ $cm^2$.

Answer:

perimeter: $11.3$ cm
area: $8$ $cm^2$