Question

in the figure below, ( overleftrightarrow{pr} ) and ( overleftrightarrow{rt} ) are perpendicular. complete the following. (a) find the slope of ( overleftrightarrow{pr} ) and the slope of ( overleftrightarrow{rt} ). slope of ( overleftrightarrow{pr} ): ( -\frac{pq}{qr} ) ( -\frac{qr}{pq} ) ( \frac{pq}{qr} ) ( \frac{qr}{pq} ) slope of ( overleftrightarrow{rt} ): ( -\frac{st}{rs} ) ( \frac{rs}{st} ) ( -\frac{rs}{st} ) ( \frac{st}{rs} ) (b) it can be shown that ( \triangle pqrsim\triangle rst ). based on this, choose the ratio that is equal to ( \frac{pq}{qr} ). ( \frac{rs}{rt} ) ( \frac{rt}{st} ) ( \frac{rs}{st} ) ( \frac{st}{rs} ) (c) using the results above, choose the correct statement below. slope of ( overleftrightarrow{pr} ) = slope of ( overleftrightarrow{rt} ) slope of ( overleftrightarrow{pr} cdot ) slope of ( overleftrightarrow{rt}=-1 ) slope of ( overleftrightarrow{pr}=- ) slope of ( overleftrightarrow{rt} ) slope of ( overleftrightarrow{pr} cdot ) slope of ( overleftrightarrow{rt}=1 ) (d) the result in part (c) is an example of the following rule for any two non - vertical perpendicular lines. the slopes of the two lines are opposites. the slopes of the two lines are reciprocals. the slopes of the two lines are the same. the slopes of the two lines are negative reciprocals.

Explanation:

Step1: Recall the slope formula

The slope of a line is \(m = \frac{\text{rise}}{\text{run}}\). For a line segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\), \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(\overleftrightarrow{PR}\), if we consider the right - triangle \(PQR\) with \(PQ\) as the vertical segment (rise) and \(QR\) as the horizontal segment (run), and since the line is decreasing (from left - to - right), the slope \(m_{PR}=-\frac{PQ}{QR}\). For \(\overleftrightarrow{RT}\), considering the right - triangle \(RST\) with \(ST\) as the vertical segment (rise) and \(RS\) as the horizontal segment (run), and since the line is increasing (from left - to - right), the slope \(m_{RT}=\frac{ST}{RS}\).

Step2: Use the similarity of triangles

Since \(\triangle PQR\sim\triangle RST\), by the property of similar triangles \(\frac{PQ}{QR}=\frac{RS}{ST}\) (corresponding sides of similar triangles are in proportion).

Step3: Calculate the product of slopes

\(m_{PR}\times m_{RT}=-\frac{PQ}{QR}\times\frac{ST}{RS}\). From \(\triangle PQR\sim\triangle RST\), \(\frac{PQ}{QR}=\frac{RS}{ST}\), so \(m_{PR}\times m_{RT}=- 1\).

Step4: Analyze the rule for perpendicular lines

If two non - vertical lines are perpendicular, and their slopes are \(m_1\) and \(m_2\), then \(m_1\times m_2=-1\). When \(m_1 =-\frac{a}{b}\) and \(m_2=\frac{b}{a}\) (where \(a,b
eq0\)), we say that \(m_1\) and \(m_2\) are negative reciprocals.

Answer:

a. slope of \(\overleftrightarrow{PR}\): \(-\frac{PQ}{QR}\); slope of \(\overleftrightarrow{RT}\): \(\frac{ST}{RS}\)
b. \(\frac{RS}{ST}\)
c. slope of \(\overleftrightarrow{PR}\cdot\) slope of \(\overleftrightarrow{RT}=-1\)
d. The slopes of the two lines are negative reciprocals.