To complete Step 5 of the proof, we need to establish that the diagonals of the quadrilateral are actually equal and use this information to state that Quadrilateral [tex]\(ABCD\)[/tex] is a rectangle, supported by the conclusion from the previous steps.
Here’s the detailed, step-by-step completion of the proof, including the statement and reason for Step 5:
```
Statement Reason
1. Quadrilateral [tex]\(ABCD\)[/tex], with [tex]\(A(1,3)\)[/tex], [tex]\(B(2,4)\)[/tex], Given
[tex]\(C(4,2)\)[/tex], [tex]\(D(3,1)\)[/tex]
2. Draw [tex]\(\overline{AC}\)[/tex] and [tex]\(\overline{BD}\)[/tex]. A line segment can be drawn between any two points.
3. [tex]\(AC = \sqrt{(4-1)^2 + (2-3)^2} Distance formula
= \sqrt{9 + 1}
\approx 3.2\)[/tex]
[tex]\(BD = \sqrt{(3-1)^2 + (1-4)^2}
= \sqrt{8 + 1}
\approx 3.2\)[/tex]
4. [tex]\(AC = BD\)[/tex] Definition of equality
5. Quadrilateral [tex]\(ABCD\)[/tex] is a rectangle. Since the lengths of the diagonals are equal, [tex]\(AC = BD\)[/tex],
and in this specific arrangement of points, this condition implies
that the quadrilateral is a rectangle.
```
Thus, the statement for Step 5 is:
"Quadrilateral [tex]\(ABCD\)[/tex] is a rectangle."
The reason for Step 5 is based on the properties of a rectangle: in a rectangle, the diagonals are congruent (equal in length). Given that the lengths of the diagonals [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are equal and applying this condition, we can conclude that Quadrilateral [tex]\(ABCD\)[/tex] is a rectangle.
