Question

this question has two parts. first, answer part a. then, answer part b. part a identify the figure with the vertices a (3,5), b (3,1), and c (0,1). triangle part b find the perimeter and area of the figure. perimeter: units area: units²

Explanation:

Step1: Calculate side - lengths 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 side $AB$ with $A(3,5)$ and $B(3,1)$:
$x_1 = 3,y_1 = 5,x_2 = 3,y_2 = 1$. Then $AB=\sqrt{(3 - 3)^2+(1 - 5)^2}=\sqrt{0+( - 4)^2}=4$.
For side $BC$ with $B(3,1)$ and $C(0,1)$:
$x_1 = 3,y_1 = 1,x_2 = 0,y_2 = 1$. Then $BC=\sqrt{(0 - 3)^2+(1 - 1)^2}=\sqrt{( - 3)^2+0}=3$.
For side $AC$ with $A(3,5)$ and $C(0,1)$:
$x_1 = 3,y_1 = 5,x_2 = 0,y_2 = 1$. Then $AC=\sqrt{(0 - 3)^2+(1 - 5)^2}=\sqrt{( - 3)^2+( - 4)^2}=\sqrt{9 + 16}=5$.

Step2: Calculate the perimeter

The perimeter $P$ of a triangle is the sum of the lengths of its sides. So $P=AB + BC+AC=4 + 3+5 = 12$.

Step3: Calculate the area

The area $A$ of a right - triangle (since $AB\perp BC$ as $x$ - coordinate of $A$ and $B$ is the same and $y$ - coordinate of $B$ and $C$ is the same) is $A=\frac{1}{2}\times base\times height$. Here, base $BC = 3$ and height $AB = 4$. So $A=\frac{1}{2}\times3\times4 = 6$.

Answer:

perimeter: 12 units
area: 6 units$^2$