Question

which statement is true?
a the slope of \\( \overline{ef} \\) is equal to the slope of \\( \overline{gh} \\) because the ratio of the vertical change to the horizontal change between points e and f is equal to the ratio of the vertical change to the horizontal change between points g and h.
b the slope of \\( \overline{ef} \\) is greater than the slope of \\( \overline{gh} \\) because the ratio of the vertical change to the horizontal change between points e and f is greater than the ratio of the vertical change to the horizontal change between points g and h.
c the slope of \\( \overline{ef} \\) is less than the slope of \\( \overline{gh} \\) because the ratio of the vertical change to the horizontal change between points e and f is less than the ratio of the vertical change to the horizontal change between points g and h.
d the slope of \\( \overline{ef} \\) is less than the slope of \\( \overline{gh} \\) because the ratio of the vertical change to the horizontal change between points e and f is less than the ratio of the vertical change to the horizontal change between points o and h.

Explanation:

Step1: Recall Slope Formula

The slope of a line segment between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{\text{vertical change}}{\text{horizontal change}}=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Analyze Slope of \(\overline{EF}\) and \(\overline{GH}\)

For two line segments, if they are parallel, their slopes are equal. From the graph, \(\overline{EF}\) and \(\overline{GH}\) appear to be parallel (or we can calculate their slopes). Let's assume we calculate the slope of \(\overline{EF}\) and \(\overline{GH}\) using the slope formula. The slope of a line is the ratio of vertical change to horizontal change. If \(\overline{EF}\) and \(\overline{GH}\) have the same slope, then the ratio of vertical change to horizontal change between their respective points should be equal. Option A states that the slope of \(\overline{EF}\) is equal to the slope of \(\overline{GH}\) because the ratio of vertical change to horizontal change between \(E\) and \(F\) is equal to that between \(G\) and \(H\), which is consistent with the slope formula. Options B, C, D suggest one slope is greater or less than the other, which would not be the case if they are parallel (or have the same slope from the graph's visual).

Answer:

A. The slope of \(\overline{EF}\) is equal to the slope of \(\overline{GH}\) because the ratio of the vertical change to the horizontal change between points \(E\) and \(F\) is equal to the ratio of the vertical change to the horizontal change between points \(G\) and \(H\).