Question

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the coordinates of the vertices of △ xyz are x(-5, 5), y(-3, -2), and z(4, 0)
drag and drop the choices into each box to correctly complete the sentences.
the slope of \\(\overline{xz}\\) is \\(\square\\), the slope of \\(\overline{yz}\\) is \\(\square\\), and the slope of \\(\overline{xy}\\) is \\(\square\\). △ xyz \\(\square\\) a right triangle because \\(\square\\)
\\( -\frac{7}{2} \\) \\( -\frac{5}{9} \\) \\( -\frac{2}{7} \\) \\( \frac{2}{7} \\) \\( \frac{9}{5} \\) is is not no two of these slopes have a product of -1 two of these slopes have a product of -1

Explanation:

Step1: Calculate slope of $\overline{XZ}$

Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$ for $X(-5,5), Z(4,0)$:
$$m_{XZ}=\frac{0-5}{4-(-5)}=\frac{-5}{9}=-\frac{5}{9}$$

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

Use slope formula for $Y(-3,-2), Z(4,0)$:
$$m_{YZ}=\frac{0-(-2)}{4-(-3)}=\frac{2}{7}$$

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

Use slope formula for $X(-5,5), Y(-3,-2)$:
$$m_{XY}=\frac{-2-5}{-3-(-5)}=\frac{-7}{2}=-\frac{7}{2}$$

Step4: Check perpendicularity

Multiply pairs of slopes:
$m_{XZ} \times m_{YZ} = -\frac{5}{9} \times \frac{2}{7}=-\frac{10}{63}
eq -1$
$m_{XZ} \times m_{XY} = -\frac{5}{9} \times -\frac{7}{2}=\frac{35}{18}
eq -1$
$m_{YZ} \times m_{XY} = \frac{2}{7} \times -\frac{7}{2}=-1$
Since two slopes have product -1, the triangle is right.

Answer:

The slope of $\overline{XZ}$ is $\boldsymbol{-\frac{5}{9}}$, the slope of $\overline{YZ}$ is $\boldsymbol{\frac{2}{7}}$, and the slope of $\overline{XY}$ is $\boldsymbol{-\frac{7}{2}}$. $\triangle XYZ$ $\boldsymbol{is}$ a right triangle because $\boldsymbol{two\ of\ these\ slopes\ have\ a\ product\ of\ -1}$