Digital SAT Math Practice Question #1171

⚡ **Visualize and Apply Key Theorems:** The fastest way to solve this problem is to quickly identify the geometric properties from the given coordinates. 1. **Identify Center and Radius:** From \((x-4)^2 + (y+2)^2 = 169\), the center is \(O(4, -2)\) and the radius is \(r=13\). 2. **Plot Relative Positions (Mentally or Sketch):** Point \(A(17, -2)\) has the same y-coordinate as \(O\), meaning \(OA\) is a horizontal radius. Point \(B(4, 11)\) has the same x-coordinate as \(O\), meaning \(OB\) is a vertical radius. 3. **Recognize the Central Angle:** Since \(OA\) is horizontal and \(OB\) is vertical, the angle \(\angle AOB\) formed at the center \(O\) is \(90^\circ\). 4. **Apply Inscribed Angle Theorem:** The inscribed angle \(\angle ACB\) subtends the same arc as the central angle \(\angle AOB\). Therefore, \(\angle ACB = \frac{1}{2} \angle AOB = \frac{1}{2} (90^\circ) = 45^\circ\). 5. **Calculate Cosine:** \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\). This method relies on quick recognition of coordinate geometry properties and the inscribed angle theorem, making it very efficient.