Question

1. the graph of line t is shown. use the similar slope triangles to compare the slopes of segment ce and segment ac.

Explanation:

Step1: Recall the slope formula

The slope formula is \(m = \frac{\text{rise}}{\text{run}}\). For a line segment with two points \((x_1,y_1)\) and \((x_2,y_2)\), \(m=\frac{y_2 - y_1}{x_2 - x_1}\). When using slope - triangles (right - triangles formed by the line segment, vertical, and horizontal lines), the slope of a line segment is the ratio of the length of the vertical side (rise) to the length of the horizontal side (run) of the slope - triangle.

Step2: Use the property of similar triangles

If two triangles are similar, the ratios of their corresponding sides are equal. Let the slope of segment \(AC\) be \(m_{AC}\) and the slope of segment \(CE\) be \(m_{CE}\).
For the slope - triangle of segment \(AC\), let the vertical side (rise) be \(AB\) and the horizontal side (run) be \(BC\). So, \(m_{AC}=\frac{AB}{BC}\).
For the slope - triangle of segment \(CE\), let the vertical side (rise) be \(CD\) and the horizontal side (run) be \(DE\).
Since the slope - triangles (the triangle with sides \(AB\) and \(BC\) and the triangle with sides \(CD\) and \(DE\)) are similar (by the AA (angle - angle) similarity criterion, as the angles formed by the line \(t\) with the horizontal and vertical lines are equal for both triangles), we know that \(\frac{AB}{BC}=\frac{CD}{DE}\).
The slope of a line segment is constant along the line. So, \(m_{AC}=\frac{\text{vertical change from }A\text{ to }C}{\text{horizontal change from }A\text{ to }C}\) and \(m_{CE}=\frac{\text{vertical change from }C\text{ to }E}{\text{horizontal change from }C\text{ to }E}\).

Answer:

The slopes of segment \(CE\) and segment \(AC\) are equal.