Final answer:
The student's question contains mixed and incomplete information, combining quadrilateral reflections and quadratic equations without a clear connection. Without specific coordinates or a coherent problem setup, it's unfeasible to determine the value of 'b' in the context of the question.
Step-by-step explanation:
The question in discussion appears to be incorrectly transcribed and combines multiple unrelated scenarios. There is no coherent arithmetic or geometric relationship established between quadrilaterals and quadratic equations in the details provided, nor are there sufficient details to address the reflection across the line y = x or the fencing of a plot of land.
For a typical reflection across the line y = x, each point (x, y) on the quadrilateral would become (y, x) after reflection. However, without specific coordinates or additional context, the value of 'b' after reflection cannot be determined.
In solving quadratic equations given by the form at² + bt + c = 0, we would use the quadratic formula: b = -b ± √(b² - 4ac) / (2a).