Question

select the correct answer.
given: \overleftrightarrow{ab} \perp \overleftrightarrow{bc}
prove: $m_{ab}m_{bc} = -1$
image of coordinate plane with points a(1,d), b(0,0), c(1,e) and lines ab, bc

statements	reasons
2. $m_{ab} = \frac{d - 0}{1 - 0} = \frac{d}{1} = d$<br>$m_{bc} = \frac{e - 0}{1 - 0} = \frac{e}{1} = e$	application of the slope formula
3. draw the vertical line segment ac	construction
4. $\angle abc$ is a right angle	definition of perpendicular
5. $\triangle abc$ is a right triangle	definition of a right triangle
6. $ba = \sqrt{1 + d^2}$<br>$bc = \sqrt{e^2 + 1}$<br>$ca = \sqrt{(d - e)^2} = d - e$	application of the distance formula
7. $(\sqrt{1 + d^2})^2 + (\sqrt{e^2 + 1})^2 = (d - e)^2$	pythagorean theorem
8. $(1 + d^2) + (e^2 + 1) = d^2 - 2de + e^2$<br>$2 + d^2 + e^2 = d^2 - 2de + e^2$<br>$2 = -2de$<br>$-1 = de$	simplify
9. $-1 = m_{ab}m_{bc}$	substitution property of equality

which step of the proof contains an error?
\bigcirc a. step 2
\bigcirc b. step 4
\bigcirc c. step 8
\bigcirc d. step 6

Explanation:

Step1: Analyze Step 6 distance calculation

The distance between $A(1,d)$ and $C(1,e)$ is $CA=\sqrt{(1-1)^2+(d-e)^2}=\sqrt{(d-e)^2}=|d-e|$, not $\sqrt{(d-e^2)}=d-e$.

Step2: Match error to correct option

This error occurs in Step 6.

Answer:

D. Step 6