To solve the problem and determine which expression is missing from step 7, we will use the Pythagorean theorem applied to right triangle [tex]\(\triangle ABC\)[/tex].
Given:
- [tex]\(\angle ABC\)[/tex] is a right angle, indicating [tex]\( \triangle ABC \)[/tex] is a right triangle.
- The side lengths are given by:
- [tex]\( BA = \sqrt{1 + d^2} \)[/tex]
- [tex]\( BC = \sqrt{e^2 + 1} \)[/tex]
- [tex]\( CA = \sqrt{(d - e)^2} = d - e \)[/tex]
The Pythagorean theorem states:
[tex]\[ AB^2 + BC^2 = AC^2 \][/tex]
So, let’s substitute the given side lengths into the Pythagorean theorem:
1. Calculate [tex]\(AB^2\)[/tex]:
[tex]\[
AB^2 = \left( \sqrt{1 + d^2} \right)^2 = 1 + d^2
\][/tex]
2. Calculate [tex]\(BC^2\)[/tex]:
[tex]\[
BC^2 = \left( \sqrt{e^2 + 1} \right)^2 = e^2 + 1
\][/tex]
3. Calculate [tex]\(AC^2\)[/tex]:
[tex]\[
AC^2 = \left( \sqrt{(d - e)^2} \right)^2 = (d - e)^2
\][/tex]
According to the Pythagorean theorem:
[tex]\[
AB^2 + BC^2 = AC^2
\][/tex]
Substitute the values we computed:
[tex]\[
(1 + d^2) + (e^2 + 1) = (d - e)^2
\][/tex]
Simplify the left-hand side:
[tex]\[
2 + d^2 + e^2 = (d - e)^2
\][/tex]
Now, expand the right-hand side:
[tex]\[
(d - e)^2 = d^2 - 2de + e^2
\][/tex]
Thus, we have:
[tex]\[
2 + d^2 + e^2 = d^2 - 2de + e^2
\][/tex]
Comparing both sides, we see that the equation:
[tex]\[
2 + d^2 + e^2 = d^2 - 2de + e^2
\][/tex]
Clearly, the expression [tex]\((d - e)^2\)[/tex] from option C matches what we need on the right-hand side of the equation.
Therefore, the missing expression is:
[tex]\[
(d - e)^2
\][/tex]
So, the correct option is C. [tex]\( (d - e)^2 \)[/tex].
