1. Question 1: Consider a line with a slope of \( \frac{3}{4} \). Draw two different slope triangles on this line. a) Explain why these triangles are similar. b) Identify the ratio of the sides of the triangles and explain how this ratio relates to the slope of the line.

To address your question, let's first understand what a slope triangle is. A slope triangle is a right triangle formed by drawing a horizontal line (run) and a vertical line (rise) from a point on a line with a given slope. The slope of a line is defined as the ratio of the rise (vertical change) to the run (horizontal change). ### a) Explanation of Similarity Two triangles are similar if they have the same shape, which means their corresponding angles are equal and their corresponding sides are in proportion. In the case of slope triangles on a line with a slope of \( \frac{3}{4} \): 1. **Angle Consistency**: Both triangles will have one angle that corresponds to the angle of the line with respect to the horizontal axis. This angle remains constant regardless of the size of the triangle, as it is determined by the slope of the line. 2. **Proportionality**: The triangles will have the same ratio of rise to run, which is \( \frac{3}{4} \). This means that if one triangle has a rise of 3 units and a run of 4 units, another triangle could have a rise of 6 units and a run of 8 units (or any other multiples of 3 and 4). Since both triangles share the same angle and the sides are proportional, they are similar by the Angle-Angle (AA) similarity criterion. ### b) Ratio of the Sides and Relation to Slope The ratio of the sides of the slope triangles can be expressed as follows: - For a triangle with a rise of 3 units and a run of 4 units, the sides are: - Vertical side (rise) = 3 - Horizontal side (run) = 4 The ratio of the sides is: \[ \text{Ratio} = \frac{\text{rise}}{\text{run}} = \frac{3}{4} \] This ratio directly corresponds to the slope of the line, which is defined as: \[ \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{3}{4} \] Thus, the ratio of the sides of the triangles (rise to run) is equal to the slope of the line. This means that the slope of the line can be interpreted as the constant ratio of vertical change to horizontal change, which is why the triangles are similar and maintain the same slope regardless of their size. In summary, the similarity of the triangles arises from their consistent angles and proportional sides, and the ratio of the sides reflects the slope of the line.