Question

in the figure below, \\( \overleftrightarrow { v p } \\) and \\( \overleftrightarrow { p z } \\) are perpendicular.

complete the following.

(a) find the slope of \\( \overleftrightarrow { v p } \\) and the slope of \\( \overleftrightarrow { p z } \\).

slope of \\( \overleftrightarrow { v p } \\): \\( \bigcirc \frac { v w } { w p } \\) \\( \bigcirc - \frac { v w } { w p } \\) \\( \bigcirc \frac { w p } { v w } \\) \\( \bigcirc - \frac { w p } { v w } \\)

slope of \\( \overleftrightarrow { p z } \\): \\( \bigcirc - \frac { p q } { q z } \\) \\( \bigcirc - \frac { q z } { p q } \\) \\( \bigcirc \frac { p q } { q z } \\) \\( \bigcirc \frac { q z } { p q } \\)

(b) it can be shown that \\( \triangle v w p \sim \triangle p q z \\).

based on this, choose the ratio that is equal to \\( \frac { v w } { w p } \\).

\\( \bigcirc \frac { p q } { p z } \\) \\( \bigcirc \frac { q z } { p q } \\) \\( \bigcirc \frac { p z } { q z } \\) \\( \bigcirc \frac { p q } { q z } \\)

(c) using the results above, choose the correct statement below.

\\( \bigcirc \\) slope of \\( \overleftrightarrow { v p } = \\) slope of \\( \overleftrightarrow { p z } \\)

\\( \bigcirc \\) slope of \\( \overleftrightarrow { v p } \cdot \\) slope of \\( \overleftrightarrow { p z } = - 1 \\)

\\( \bigcirc \\) slope of \\( \overleftrightarrow { v p } \cdot \\) slope of \\( \overleftrightarrow { p z } = 1 \\)

\\( \bigcirc \\) slope of \\( \overleftrightarrow { v p } = - \\) slope of \\( \overleftrightarrow { p z } \\)

(d) the result in part (c) is an example of the following rule for any two non - vertical perpendicular lines.

\\( \bigcirc \\) the slopes of the two lines are opposites.

\\( \bigcirc \\) the slopes of the two lines are reciprocals.

\\( \bigcirc \\) the slopes of the two lines are negative reciprocals.

\\( \bigcirc \\) the slopes of the two lines are the same.

Explanation:

(a)

Step1: Recall the slope formula

The slope of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For the line \(\overleftrightarrow{VP}\), if we consider the right - triangle \(VWP\) (where \(V\) and \(P\) are two points on the line \(\overleftrightarrow{VP}\)), and assume \(V=(x_1,y_1)\) and \(P=(x_2,y_2)\) with \(VW\) as the vertical change (\(y_2 - y_1\)) and \(WP\) as the horizontal change (\(x_2 - x_1\)). The slope of \(\overleftrightarrow{VP}\) is \(m_{VP}=\frac{VW}{WP}\) (since slope \(m=\frac{\text{rise}}{\text{run}}\)).
For the line \(\overleftrightarrow{PZ}\), considering the right - triangle \(PQZ\) (where \(P\) and \(Z\) are two points on the line \(\overleftrightarrow{PZ}\)), with \(QZ\) as the vertical change (negative in the direction of the slope calculation) and \(PQ\) as the horizontal change. The slope of \(\overleftrightarrow{PZ}\) is \(m_{PZ}=-\frac{QZ}{PQ}\) (because the line is decreasing, so the slope is negative and \(m =-\frac{\text{rise}}{\text{run}}\)).

(b)

Step1: Use the property of similar triangles

Since \(\triangle VWP\sim\triangle PQZ\), by the property of similar triangles \(\frac{VW}{WP}=\frac{PQ}{QZ}\) (corresponding sides of similar triangles are in proportion).

(c)

Step1: Calculate the product of slopes

We know \(m_{VP}=\frac{VW}{WP}\) and \(m_{PZ}=-\frac{QZ}{PQ}\). From \(\frac{VW}{WP}=\frac{PQ}{QZ}\) (from part (b)), then \(m_{VP}\times m_{PZ}=\frac{VW}{WP}\times(-\frac{QZ}{PQ})=- 1\).

(d)

Step1: Recall the slope rule for perpendicular lines

If two non - vertical lines are perpendicular, the product of their slopes is \(-1\). If \(m_1\) and \(m_2\) are the slopes of two perpendicular lines, \(m_1 =-\frac{1}{m_2}\), which means the slopes are negative reciprocals.

Answer:

(a) slope of \(\overleftrightarrow{VP}\): \(\frac{VW}{WP}\); slope of \(\overleftrightarrow{PZ}\): \(-\frac{QZ}{PQ}\)
(b) \(\frac{PQ}{QZ}\)
(c) slope of \(\overleftrightarrow{VP}\cdot\) slope of \(\overleftrightarrow{PZ}=-1\)
(d) The slopes of the two lines are negative reciprocals.