The problem you're asking about involves a geometric configuration with three lines and the area of a triangle formed by their intersections. Here's a breakdown of the question:

• We are given two lines:

  1. $l_1: y = 2x$
  2. $l_2: y = \frac{3}{4}x + \frac{5}{2}$

• A third line, $l_3$, is the result of rotating line $l_1$ by 90° clockwise around the point $(0, 0)$.

• The intersections of these lines happen at points A, B, and C. The problem asks to find the area of triangle $ABC$, denoted by $[ABC]$, using the area of a triangle formed by points X, Y, and Z, denoted by $[XYZ]$, on the rotated plane.

Step-by-step Plan:

1. Find the intersection points of the given lines:

   • First, calculate the intersection of lines $l_1$ and $l_2$ (this gives point A).
   • Then, find the intersection of line $l_1$ and line $l_3$ (this gives point B).
   • Finally, find the intersection of line $l_2$ and line $l_3$ (this gives point C).

2. Rotate line $l_1$ by 90° around $(0, 0)$. This can be done by applying the standard rotation transformation matrix to the equation of $l_1$.

3. Find the area of triangle $ABC$ formed by the points of intersection.

Would you like me to guide you through these steps one by one, or would you prefer I proceed with all the calculations at once?