Question

which of these can be used to show that the slope of a non - vertical line is the same anywhere along the line?
show hints
acute triangles
obtuse triangles
right triangles
any triangles

Explanation:

Step1: Recall the slope formula

The slope \(m\) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Geometrically, if we consider two right - angled triangles formed by the line, the vertical side (rise) and the horizontal side (run) of the right - triangle correspond to \(y_2 - y_1\) and \(x_2 - x_1\) respectively.

Step2: Use similar right - triangles property

When we take two different pairs of points on the non - vertical line, the right - triangles formed (by dropping perpendiculars from the points to the \(x\) and \(y\) axes) are similar. For similar right - triangles, the ratios of their corresponding sides are equal. The ratio of the vertical side to the horizontal side (which is the slope of the line) is the same for all such right - triangles formed by the line. Acute triangles (\(0^{\circ}<\text{angles}<90^{\circ}\)) and obtuse triangles (one angle \(> 90^{\circ}\)) do not have the property that their side - ratios (corresponding to the slope concept) are easily related in the context of a straight line. And not all triangles (any triangles) can be used as the slope is related to the ratio of vertical and horizontal changes which is best modeled by right - triangles.

Answer:

C. right triangles