Let's delve into finding the missing expression in step 7 using clear math steps.
Given:
[tex]\[ BA = \sqrt{1 + d^2} \][/tex]
[tex]\[ BC = \sqrt{e^2 + 1} \][/tex]
We're applying the distance formula:
[tex]\[ CA = \sqrt{(d - e)^2} = d - e \][/tex]
(Note: There seems to be a typo here, as [tex]\(\sqrt{(d - e)^2}\)[/tex] simplifies to [tex]\(|d - e|\)[/tex]. However, this can be overlooked since the primary focus is on the provided expressions.)
We need to simplify:
[tex]\[ \left(\sqrt{1 + d^2}\right)^2 + \left(\sqrt{e^2 + 1}\right)^2 \][/tex]
Step-by-step:
1. Square both expressions:
[tex]\[
(\sqrt{1 + d^2})^2 = 1 + d^2
\][/tex]
[tex]\[
(\sqrt{e^2 + 1})^2 = e^2 + 1
\][/tex]
2. Add the squared results:
[tex]\[
(1 + d^2) + (e^2 + 1)
\][/tex]
This simplifies to:
[tex]\[
1 + d^2 + e^2 + 1 = d^2 + e^2 + 2
\][/tex]
Looking at the subsequent lines:
[tex]\[ 2 + d^2 + e^2 = d^2 - 2de + e^2 \][/tex]
Let's simplify the right-hand side of the equation using a known algebraic identity. Notice:
[tex]\[ d^2 - 2de + e^2 \][/tex]
This represents the expansion of:
[tex]\[
(d - e)^2
\][/tex]
So, comparing the left-hand side and right-hand side of the equation:
[tex]\[ d^2 + e^2 + 2 \text{ (From: } (\sqrt{1+d^2})^2+(\sqrt{e^2+1})^2) = d^2 - 2 d e + e^2 \][/tex]
To find the missing term that equates both sides, we examine:
[tex]\[ 2 = -2de \][/tex]
Thus,
[tex]\[ \boxed{-2 d e} \][/tex]
Therefore, the missing expression in step 7 is:
A. [tex]\(-2 d e\)[/tex]
This ensures the equation balance, verifying that option A is the correct missing expression.
