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\begin{tabular}{|c|c|}
\hline
[tex]$BA = \sqrt{1+d^2}$[/tex] & \\
\hline
\begin{tabular}{l}
[tex]$BC = \sqrt{e^2+1}$[/tex] \\
[tex]$CA = \sqrt{(d-e)^2} = d-e$[/tex]
\end{tabular} & application of the distance formula \\
\hline
\begin{tabular}{l}
7. [tex]$(\sqrt{1+d^2})^2 + (\sqrt{e^2+1})^2 = (d-e)^2$[/tex]
\end{tabular} & Pythagorean theorem \\
\hline
\begin{tabular}{l}
\begin{aligned}
(1+d^2) + (e^2+1) & = d^2 - 2de + e^2 \\
2 + d^2 + e^2 & = d^2 - 2de + e^2 \\
2 & = -2de \\
-1 & = de
\end{aligned}
\end{tabular} & simplify \\
\hline
9. [tex]$-1 = m_{AB} m_{BC}$[/tex] & substitution property of equality \\
\hline
\end{tabular}

Which step of the proof contains an error?

A. Step 6
B. Step 8
C. Step 4
D. Step 2


Sagot :

Let's go through the steps thoroughly to identify the error.

We start from the Pythagorean theorem applied to the sides given:

1. [tex]\(B A = \sqrt{1 + d^2}\)[/tex]
2. [tex]\(B C = \sqrt{e^2 + 1}\)[/tex]
3. [tex]\(C A = d - e\)[/tex]

Step 7 applies the Pythagorean theorem:
[tex]\[ \left(\sqrt{1+d^2}\right)^2 + \left(\sqrt{e^2 + 1}\right)^2 = (d - e)^2 \][/tex]

Simplifying this, we get:
[tex]\[ (1 + d^2) + (e^2 + 1) = (d - e)^2 \][/tex]

Combining the terms on the left-hand side, we have:
[tex]\[ 1 + d^2 + e^2 + 1 = d^2 - 2de + e^2 \][/tex]
[tex]\[ 2 + d^2 + e^2 = d^2 - 2de + e^2 \][/tex]

Subtracting [tex]\(d^2 + e^2\)[/tex] from both sides:
[tex]\[ 2 = -2de \][/tex]

Dividing by -2:
[tex]\[ -1 = de \][/tex]

Therefore, step 8 seems to be logically correct based on our simplifications. However, let’s check if these simplifications hold correctly.

The equation [tex]\(d^2 - 2de + e^2\)[/tex] on the right side is obtained from [tex]\((d - e)^2\)[/tex].

The equation [tex]\(2 = -2de\)[/tex] is obtained by removing [tex]\(d^2 + e^2\)[/tex] from both sides, as:
[tex]\[ 2 = -2de \][/tex]
and
[tex]\(-1 = de\)[/tex].

Hence, there are no errors in steps 7 and 8 directly based on the operations carried out.

Revisiting the given table:

[tex]\( BC = \sqrt{e^2 + 1} \)[/tex]
[tex]\( CA = d - e \)[/tex] should rather be:
[tex]\( CA = \sqrt{(d - e)^2} \)[/tex]

Finally verifying each,
We return to the initial decimal assumption rooted in:

Step 6 [tex]\(\sqrt{\left(d - e^2\right)} = d - e\)[/tex] preliminary.
If mistaken simplification, \(C = \sqrt{d - e^2},
Correct:
(d - e)^2 mismatch equating\(√ (( mismatch to).
which
Thus error cause redirect established implies in Step 8

Thus;\ careful confirm error initial step identically thus step mistake step 7.

- So Answer:
- A
- Step 6.