Question

in the figure below, \\( \overleftrightarrow { e f } \\) and \\( \overleftrightarrow { g h } \\) are parallel.

complete the following.
(a) find the slope of \\( \overleftrightarrow { e f } \\) and the slope of \\( \overleftrightarrow { g h } \\).
slope of \\( \overleftrightarrow { e f } \\): \\( \bigcirc \frac { i f } { i e } \\) \\( \bigcirc - \frac { i e } { i f } \\) \\( \bigcirc \frac { i e } { i f } \\) \\( \bigcirc - \frac { i f } { i e } \\)
slope of \\( \overleftrightarrow { g h } \\): \\( \bigcirc \frac { i g } { i h } \\) \\( \bigcirc - \frac { i h } { i g } \\) \\( \bigcirc - \frac { i g } { i h } \\) \\( \bigcirc \frac { i h } { i g } \\)
(b) it can be shown that \\( \triangle i e f \sim \triangle i g h \\).
based on this, choose the ratio that is equal to \\( \frac { i e } { i f } \\).
\\( \bigcirc \frac { i g } { g h } \\) \\( \bigcirc \frac { i g } { i h } \\) \\( \bigcirc \frac { i h } { i g } \\) \\( \bigcirc \frac { g h } { i h } \\)
(c) using the results above, choose the correct statement below.
\\( \bigcirc \\) slope of \\( \overleftrightarrow { e f } \cdot \\) slope of \\( \overleftrightarrow { g h } = 1 \\)
\\( \bigcirc \\) slope of \\( \overleftrightarrow { e f } = - \\) slope of \\( \overleftrightarrow { g h } \\)
\\( \bigcirc \\) slope of \\( \overleftrightarrow { e f } \cdot \\) slope of \\( \overleftrightarrow { g h } = - 1 \\)
\\( \bigcirc \\) slope of \\( \overleftrightarrow { e f } = \\) slope of \\( \overleftrightarrow { g h } \\)
(d) the result in part (c) is an example of the following rule for any two non - vertical parallel lines.
\\( \bigcirc \\) the slopes of the two lines are the same.
\\( \bigcirc \\) the slopes of the two lines are negative reciprocals.
\\( \bigcirc \\) the slopes of the two lines are opposites.
\\( \bigcirc \\) the slopes of the two lines are reciprocals.

Explanation:

Step1: Recall the slope formula

The slope of a line is \(m=\frac{\text{rise}}{\text{run}}\). For line \(\overleftrightarrow{EF}\), the "rise" is \(IE\) and the "run" is \(IF\), so slope of \(\overleftrightarrow{EF}=\frac{IE}{IF}\). For line \(\overleftrightarrow{GH}\), the "rise" is \(IG\) and the "run" is \(IH\), so slope of \(\overleftrightarrow{GH}=\frac{IG}{IH}\).

Step2: Use the property of similar triangles

Since \(\triangle IEF\sim\triangle IGH\), we have \(\frac{IE}{IF}=\frac{IG}{IH}\) (corresponding sides of similar triangles are in proportion).

Step3: Analyze the slopes of parallel lines

If two lines are parallel, their slopes are equal. Let \(m_1 = \frac{IE}{IF}\) (slope of \(\overleftrightarrow{EF}\)) and \(m_2=\frac{IG}{IH}\). From \(\triangle IEF\sim\triangle IGH\), \(m_1 = m_2\).

Answer:

a. slope of \(\overleftrightarrow{EF}\): \(\frac{IE}{IF}\); slope of \(\overleftrightarrow{GH}\): \(\frac{IG}{IH}\)
b. \(\frac{IG}{IH}\)
c. slope of \(\overleftrightarrow{EF}=\) slope of \(\overleftrightarrow{GH}\)
d. The slopes of the two lines are the same.