Question

lydia graphed $\triangle lmn$ at the coordinates $l (0, 0)$, $m (2, 2)$, and $n (2, -1)$. she thinks $\triangle lmn$ is a right triangle. is lydias assertion correct?\
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$\bigcirc$ yes; the slopes of $\overline{lm}$ and $\overline{ln}$ are opposite reciprocals.\
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$\bigcirc$ no; the slopes of $\overline{lm}$ and $\overline{ln}$ are not opposite reciprocals.\
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$\bigcirc$ yes; the slopes of $\overline{lm}$ and $\overline{ln}$ are the same.\
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$\bigcirc$ no; the slopes of $\overline{lm}$ and $\overline{ln}$ are not the same.

Explanation:

Step1: Recall slope formula

Slope between 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 $\overline{LM}$

Points $L(0,0)$ and $M(2,2)$:
$m_{LM}=\frac{2-0}{2-0}=\frac{2}{2}=1$

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

Points $L(0,0)$ and $N(2,-1)$:
$m_{LN}=\frac{-1-0}{2-0}=-\frac{1}{2}$

Step4: Check perpendicularity

Two lines are perpendicular if slopes are opposite reciprocals (product = -1).
$m_{LM} \times m_{LN} = 1 \times (-\frac{1}{2}) = -\frac{1}{2}
eq -1$
Wait, correct check: actually, check $\overline{MN}$ and $\overline{LN}$ or $\overline{LM}$? Wait, recalculate slope of $\overline{MN}$: points $M(2,2)$ and $N(2,-1)$:
$m_{MN}=\frac{-1-2}{2-2}=\frac{-3}{0}$ (undefined, vertical line)
Slope of $\overline{LN}$ is $-\frac{1}{2}$, slope of $\overline{LM}$ is 1. Wait, vertical line $\overline{MN}$ is perpendicular to horizontal line, but $\overline{LM}$ is not horizontal. Wait, no: the correct pair is $\overline{MN}$ (vertical) and... no, wait, the options only reference $\overline{LM}$ and $\overline{LN}$. Their product is $1 \times (-\frac{1}{2}) = -\frac{1}{2}
eq -1$, so they are not opposite reciprocals. Thus the triangle is not right based on these two sides, and Lydia is wrong.

Answer:

No; the slopes of $\overline{LM}$ and $\overline{LN}$ are not opposite reciprocals.