Use similar triangles to determine the slope of the line. Write your answer in fraction form.
To determine the slope of a line using similar triangles, we need to find the ratio of the vertical change to the horizontal change on the line.
Let's consider two points on the line, $A$ and $B$, with coordinates $(x_1,y_1)$ and $(x_2,y_2)$ respectively.
The slope of the line is equal to the ratio of the vertical change to the horizontal change between these two points:
$$\text{slope} = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{y_2-y_1}{x_2-x_1}.$$
Therefore, the slope of the line in fraction form is $\boxed{\frac{y_2-y_1}{x_2-x_1}}$.
To use similar triangles to determine the slope of a line, we can compare the corresponding side lengths of two similar triangles.
Let's consider two similar triangles: triangle ABC and triangle DEF. The corresponding sides are AB and DE, BC and EF, and AC and DF.
The slope of a line can be found by calculating the ratio of the vertical change (change in y-coordinates) to the horizontal change (change in x-coordinates) between two points on the line.
Let's choose two corresponding points on the line represented by triangle ABC and DEF. Let's say point A corresponds to point D, and point B corresponds to point E.
The vertical change between points A and B is the difference in their y-coordinates: Δy = yB - yA.
The horizontal change between points A and B is the difference in their x-coordinates: Δx = xB - xA.
Similarly, the vertical change between points D and E is Δy' = yE - yD, and the horizontal change between points D and E is Δx' = xE - xD.
Since the triangles ABC and DEF are similar, the corresponding side lengths are in proportion. This means that the ratio of the vertical changes is equal to the ratio of the horizontal changes:
Δy/Δx = Δy'/Δx'
To find the slope in fraction form, we can simplify this ratio by dividing both Δy and Δx by their greatest common divisor (gcd). This will give us the slope in the form of a fraction.
So, the slope of the line, as determined using similar triangles, is Δy/Δx in fraction form.