Question

statements reasons 1. r || s given 2. $m_{r}=\frac{d - b}{c - 0}=\frac{d - b}{c}$ $m_{s}=\frac{0 - a}{c - 0}=-\frac{a}{c}$ application of the slope formula 3. distance from $(0,b)$ to $(0,a)$ equals the distance from $(c,d)$ to $(c,0)$ definition of parallel lines 4. $d - 0=b - a$ application of the distance formula 5. $m_{r}=\frac{(b - a)-b}{c}$ substitution property of equality 6. $m_{r}=\frac{a}{c}$ inverse property of addition 7. $m_{r}=m_{s}$ substitution property of equality which step of the proof contains an error? a. step 4 b. step 2 c. step 5 d. step 6

Explanation:

Step1: Analyze Step 2

The slope - formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ is correctly applied to find the slopes of lines $r$ and $s$. So, Step 2 is correct.

Step2: Analyze Step 4

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $(0,b)$ and $(0,a)$, the distance is $|b - a|$, and for points $(c,d)$ and $(c,0)$ the distance is $|d-0|$. Just setting $d - 0=b - a$ is incorrect because we should consider the absolute - value. This step contains an error.

Step3: Analyze Step 5

If we assume the incorrect result from Step 4, the substitution in Step 5 is a correct application of the substitution property of equality. But since Step 4 is wrong, Step 5 is based on a wrong premise, yet the operation within Step 5 (substitution) is correct in itself.

Step4: Analyze Step 6

The simplification in Step 6 using the inverse property of addition is correct if we consider the expression from Step 5.

Answer:

A. Step 4