Question

lesson 3 - 3 exit ticket for items 1 and 2, use the graph to complete the exercises. 1. write the ratio of side lengths for each right triangle. 2. what are the slopes of the hypotenuses? how does this relate to the slope of the line? name__ date period lesson 3 - 4 exit ticket for items 1 and 2, use the graph showing the average distance max runs per hour to complete the exercises. 1. consider a right triangle whose hypotenuse lies on the line. complete the sentence. the ratio of the height of any right triangle drawn to the base of any right triangle is to__ 2. what is the equation of the line in the form y = mx for the graph?

Explanation:

Lesson 3 - 3

1.

Step1: Identify right - triangle sides

Let's assume two points on the line to form right - triangles. For example, if we take two points \((x_1,y_1)\) and \((x_2,y_2)\) on the line. The vertical side (rise) and horizontal side (run) of the right - triangle are used to form the ratio. If we consider two consecutive grid - points, say \((0,0)\) and \((1,1)\), the ratio of the side lengths of the right - triangle (rise to run) is \(1:1\). In general, for any two points \((x_1,y_1)\) and \((x_2,y_2)\) on the line, the ratio of the vertical side length to the horizontal side length is \(\frac{y_2 - y_1}{x_2 - x_1}\).

Step1: Recall slope formula

The slope \(m\) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\). The slope of the hypotenuse of a right - triangle formed by two points on the line is the same as the slope of the line because the line is a straight line and the ratio of vertical change to horizontal change (rise over run) is constant for any two points on the line.

Step2: Calculate slope

If we take two points \((0,0)\) and \((1,1)\) on the line, \(m=\frac{1 - 0}{1 - 0}=1\). The slopes of all hypotenuses of right - triangles formed on the line are equal to the slope of the line because the line has a constant rate of change.

Step1: Analyze the right - triangle on the line

For a right - triangle whose hypotenuse lies on the line representing the running distance - time graph, the ratio of the height (vertical side, change in distance) of any right - triangle drawn to the base (horizontal side, change in time) is equal to the speed. Since the line is a straight line, this ratio is constant. If we take two points \((1,5)\) and \((2,10)\) on the line, the ratio of the height to the base is \(\frac{10 - 5}{2 - 1}=5\).

Answer:

The ratio of side lengths of the right - triangle is \(\frac{\text{rise}}{\text{run}}\) (e.g., if points are \((0,0)\) and \((1,1)\), ratio is \(1:1\))

2.