Question

consider the graph shown. which of the following does not represent the rate of change found when using similar triangles? select all that apply.
a. $\frac{4}{6}$
b. $\frac{2}{3}$
c. $-\frac{4}{6}$
d. $-\frac{2}{3}$

Explanation:

Step1: Determine the slope formula

The slope (rate of change) formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For similar triangles on a line, the slope is constant. Let's take two points. For example, if we take \(C(0,0)\) and \(D(- 6,4)\), then \(m=\frac{4 - 0}{-6-0}=-\frac{2}{3}\). If we take \(C(0,0)\) and \(A(12,-8)\), then \(m=\frac{-8 - 0}{12 - 0}=-\frac{2}{3}\). Also, \(-\frac{4}{6}=-\frac{2}{3}\) (by simplifying the fraction).

Step2: Analyze each option

• Option a: \(\frac{4}{6}=\frac{2}{3}\). Since the slope (rate of change) of the line is negative (\(m =-\frac{2}{3}\)), \(\frac{4}{6}\) does not represent the rate of change.
• Option b: \(\frac{2}{3}\). Since the slope (rate of change) of the line is negative (\(m =-\frac{2}{3}\)), \(\frac{2}{3}\) does not represent the rate of change.
• Option c: \(-\frac{4}{6}=-\frac{2}{3}\), which is the slope (rate of change) of the line.
• Option d: \(-\frac{2}{3}\), which is the slope (rate of change) of the line.

Answer:

A. \(\frac{4}{6}\), B. \(\frac{2}{3}\)