Question

a line and two triangles are shown on the coordinate plane.
which statements are true? select all that apply.
the slope of (overline{k}) is equal to the slope of (overline{kl}).
the slope of (overline{j}) is equal to the slope of (overline{ml}).
the slope of (overline{kl}) is equal to the slope of the line.
the slope of the line is equal to (\frac{ml}{mm}).
the slope of the line is equal to (\frac{mm}{ml}).
the slope of the line is equal to (\frac{k}{ak}).

Explanation:

Step1: Recall the slope formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Analyze the slope of the line

For any two points on the same line, the slope is the same. Triangles \( \triangle HKL\) and \( \triangle MNL\) are similar (by AA similarity, as the angles are equal because of the parallel sides - vertical and horizontal sides of the right - angled triangles).
The slope of the line can be calculated using the vertical and horizontal sides of the right - angled triangles.
For a right - angled triangle with vertical side (rise) and horizontal side (run), the slope \(m=\frac{\text{rise}}{\text{run}}\).
Let's assume for \( \triangle HKL\): if \(HK\) is the vertical side and \(KL\) is the horizontal side, the slope of the line (using the points on the line) is \(m = \frac{HK}{KL}\). For \( \triangle MNL\), if \(MN\) is the vertical side and \(NL\) is the horizontal side, the slope of the line is also \(m=\frac{MN}{NL}\).
Since the line has a constant slope, the slope of \( \overline{HK}\) (calculated as part of the line's slope using \( \triangle HKL\)) is equal to the slope of \( \overline{MN}\) (calculated as part of the line's slope using \( \triangle MNL\)). Also, the slope of \( \overline{KL}\) (as part of the line's slope calculation) is equal to the slope of the line (because \(m=\frac{y_2 - y_1}{x_2 - x_1}\) for the line, and for the right - angled triangle on the line \(m=\frac{\text{vertical change (related to }HK\text{ or }MN\text{)}}{\text{horizontal change (related to }KL\text{ or }NL\text{)}}\)).

Answer:

The slope of \( \overline{HK}\) is equal to the slope of \( \overline{MN}\), The slope of \( \overline{KL}\) is equal to the slope of the line, The slope of the line is equal to \(\frac{MN}{NL}\)