Question

the coordinates of the vertices of △ghj are g(-1, 3), h(1, 2), and j(-3, -1). drag and drop the choices into each box to correctly complete the sentences. the slope of $overline{gh}$ is, the slope of $overline{hj}$ is, and the slope of $overline{jg}$ is. △ghj a right triangle because no two of these slopes have a product of -1. -$\frac{4}{3}$, -$\frac{1}{2}$, $\frac{1}{2}$, $\frac{3}{4}$, 2, is, is not

Explanation:

Step1: Recall slope - formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of $\overline{GH}$

For points $G(-1,3)$ and $H(1,2)$, $x_1=-1,y_1 = 3,x_2=1,y_2 = 2$. Then $m_{GH}=\frac{2 - 3}{1-(-1)}=\frac{-1}{2}=-\frac{1}{2}$.

Step3: Calculate slope of $\overline{HJ}$

For points $H(1,2)$ and $J(-3,-1)$, $x_1 = 1,y_1=2,x_2=-3,y_2=-1$. Then $m_{HJ}=\frac{-1 - 2}{-3 - 1}=\frac{-3}{-4}=\frac{3}{4}$.

Step4: Calculate slope of $\overline{JG}$

For points $J(-3,-1)$ and $G(-1,3)$, $x_1=-3,y_1=-1,x_2=-1,y_2 = 3$. Then $m_{JG}=\frac{3-(-1)}{-1-(-3)}=\frac{4}{2}=2$.

Step5: Check if it's a right - triangle

Two lines with slopes $m_1$ and $m_2$ are perpendicular if $m_1\times m_2=-1$. Since $-\frac{1}{2}\times\frac{3}{4}
eq - 1$, $-\frac{1}{2}\times2
eq - 1$, and $\frac{3}{4}\times2
eq - 1$, no two of these slopes have a product of $-1$. So $\triangle GHJ$ is not a right - triangle.

Answer:

The slope of $\overline{GH}$ is $-\frac{1}{2}$, the slope of $\overline{HJ}$ is $\frac{3}{4}$, the slope of $\overline{JG}$ is $2$, $\triangle GHJ$ is not a right triangle because no two of these slopes have a product of $-1$.