In the technical process of graphing a hyperbola, a 'central rectangle' is constructed using the vertices and conjugate axis endpoints. What geometric feature of the hyperbola is established by extending the diagonals of this rectangle?

Hyperbolic Design Framework

In technical drafting and architectural layout, the central rectangle of a hyperbola is a critical construction guide. Match each component of the rectangle with its specific role in the graphing process.

In technical drafting and design, a 'central rectangle' is often constructed to assist in graphing a hyperbola centered at ${}0, 0$. Is it true or false that the sides of this rectangle are designed to intersect the coordinate axes at the hyperbola's vertices $(\pm a, 0)$ and the points $(0, \pm b)$?

You are updating a technical drafting manual for an engineering firm that designs hyperbolic cooling towers. Complete the following guideline for the new design trainees:

"To accurately sketch the boundaries of the curve, first construct the central rectangle using the vertices and the conjugate axis endpoints. The extended diagonals of this central rectangle form the ________ of the hyperbola, which guide the precise shape of the outward curves."