triangle mno is congruent to right triangle rst with a right angle at vertex r. if the slope of $ \overline{rs} $ is $ - \frac { 1 } { 5 } $, what must be true?
the slope of $ \overline { t r } $ is 5.
the slope of $ \overline { o m } $ is 5.
the slope of $ \overline { m n } $ is $ - \frac { 1 } { 5 } $
the slope of $ \overline { n o } $ is $ - \frac { 1 } { 5 } $

Question

Accepted Answer

# Explanation:
## Step1: Recall the property of perpendicular lines
If two lines are perpendicular, the product of their slopes is $- 1$. Let the slope of one line be $m_1$ and the slope of the other line be $m_2$, then $m_1\times m_2=-1$.
Given that the slope of $\overline{RS}$ is $m_{RS}=-\frac{1}{5}$, and $\angle R = 90^{\circ}$, so $\overline{RS}\perp\overline{RT}$. Let the slope of $\overline{RT}$ be $m_{RT}$.
Using the formula $m_{RS}\times m_{RT}=-1$, substitute $m_{RS}=-\frac{1}{5}$ into it: $-\frac{1}{5}\times m_{RT}=-1$.
## Step2: Solve for $m_{RT}$
Solve the equation $-\frac{1}{5}\times m_{RT}=-1$ for $m_{RT}$. Multiply both sides of the equation by $- 5$: $m_{RT}=5$.

Since $\triangle MNO\cong\triangle RST$, corresponding sides and angles are congruent. But the key here is the relationship between the slopes of perpendicular lines in $\triangle RST$ which is transferred to the properties of the slopes in $\triangle MNO$ through congruence. The first option about the slope of $\overline{TR}$ (which is the same as the slope of $\overline{RT}$) is correct based on the perpendicular - slope relationship.

# Answer:
The slope of $\overline{TR}$ is $5$.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the property of perpendicular lines

Step2: Solve for \(m_{RT}\)

Answer: