3. Let ABCD be a square and P be a point on segment AB. Segments CP and BD intersect at Q, and BD is extended beyond B to a point S. Finally, T is the intersection of line CP and the line through S parallel to AB. If AB = 18 and PT = 2CQ, what is the area of quadrilateral PST?

To solve this problem, we proceed with the following steps:

### Step 1: Assign coordinates to the square
Let the square \( ABCD \) be on the coordinate plane with:
- \( A = (0, 0) \),
- \( B = (18, 0) \),
- \( C = (18, 18) \),
- \( D = (0, 18) \).

Let \( P = (x, 0) \) on segment \( AB \), where \( 0 \leq x \leq 18 \).

### Step 2: Equations of lines
- Line \( CP \): Slope is \( \frac{18}{x - 18} \), so the equation of \( CP \) is:
  \[
  y - 0 = \frac{18}{x - 18} (x - 18) \quad \implies \quad y = \frac{18}{x - 18} (X - x).
  \]



Let ABCD be a square and P be a point on segment AB. Segments CP and BD intersect at Q, and BD is extended beyond B to a point S. Finally, T is the intersection of line CP and the line through S parallel to AB. If AB = 18 and PT = 2CQ, what is the area of quadrilateral PST?

Explore this challenging geometry problem involving a square ABCD, intersecting lines CP and BD, and the area of quadrilateral PST. This detailed solution uses coordinate geometry and line equations to find the area. Suitable for high school students in Grades 10-12.

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