Question

this question has two parts. first, answer part a. then, answer part b. part a identify the figure with the vertices g(-4,1), h(4,1), and i(0, - 2). triangle part b find the perimeter and area of the figure. perimeter units area units²

Explanation:

Step1: Calculate the lengths of the sides using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$

For side $GH$ with $G(-4,1)$ and $H(4,1)$:
$d_{GH}=\sqrt{(4 - (-4))^2+(1 - 1)^2}=\sqrt{(8)^2+0^2}=8$
For side $GI$ with $G(-4,1)$ and $I(0,-2)$:
$d_{GI}=\sqrt{(0 - (-4))^2+(-2 - 1)^2}=\sqrt{(4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$
For side $HI$ with $H(4,1)$ and $I(0,-2)$:
$d_{HI}=\sqrt{(0 - 4)^2+(-2 - 1)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16+9}=\sqrt{25}=5$

Step2: Calculate the perimeter

Perimeter $P=d_{GH}+d_{GI}+d_{HI}=8 + 5+5=18$

Step3: Calculate the area

Since $GI = HI = 5$ and $GH = 8$, this is an isosceles triangle. The base is $GH = 8$ and the height $h$ from $I$ to $GH$ can be found as the vertical distance from $y=-2$ to $y = 1$, so $h=3$.
Area $A=\frac{1}{2}\times base\times height=\frac{1}{2}\times8\times3 = 12$

Answer:

perimeter: 18 units
area: 12 units$^2$