Digital SAT Math Practice Question #1197

⚡ **Geometric Insight / Symmetry:** 1. **Find the center quickly:** Points A(0,0) and B(6,0) are on the x-axis. The perpendicular bisector of AB is the line \(x = (0+6)/2 = 3\). Since the center of the circle lies on \(y=x\), the center must be at the intersection of \(x=3\) and \(y=x\), which is \(C(3,3)\). 2. **Observe the lines PA and PB:** * Point P is (6,6). * Point A is (0,0). The line segment PA connects (0,0) to (6,6), which is the line \(y=x\). This line makes an angle of \(45^\circ\) with the positive x-axis. * Point B is (6,0). The line segment PB connects (6,0) to (6,6). This is a vertical line \(x=6\). 3. **Calculate the angle:** The angle \(\angle APB\) is the angle between the line \(y=x\) (line PA) and the vertical line \(x=6\) (line PB). A vertical line makes an angle of \(90^\circ\) with the x-axis. The line \(y=x\) makes an angle of \(45^\circ\) with the x-axis. The angle between a line with slope 1 and a vertical line is \(45^\circ\). You can visualize this by drawing a horizontal line through P; the angle between PA and this horizontal line is \(45^\circ\), and PB is perpendicular to this horizontal line. Thus, \(\angle APB = 45^\circ\). This method leverages coordinate geometry observations to quickly deduce the angle without explicit circle calculations.