This question involves a circle tangent to the x-axis. When a circle with center (4, 1) is tangent to the x-axis, the radius equals the y-coordinate of the center (the perpendicular distance to the x-axis). Therefore, r = 1. The equation is (x-4)² + (y-1)² = 1² = 1. A common mistake is using the x-coordinate as the radius when tangent to the x-axis. Remember: tangent to x-axis means r = |y-coordinate of center|.

This question requires finding area given circumference. From C = 2πr = 30π, we get r = 15 cm. The area is A = πr² = π(15)² = 225π square centimeters. A common error is using the circumference value directly without first finding the radius. Always work through the radius: circumference → radius → area.

This question asks for the equation of a circle centered at the origin passing through (0, -9). Since the center is (0, 0) and the circle passes through (0, -9), the radius is the distance from origin to (0, -9), which is 9. The equation is x² + y² = r² = 81. A common error is using the y-coordinate (-9) as the radius without taking its absolute value. Remember that radius is always positive, representing distance.

This question involves a semicircle with diameter AB and point C on the circle. When a triangle is inscribed in a semicircle with one side as the diameter, the angle at C is 90°. Given AC = 12 and BC = 16, we can find AB using the Pythagorean theorem: AB² = AC² + BC² = 144 + 256 = 400, so AB = 20. Since AB is the diameter, radius = 20/2 = 10. The key insight is recognizing that angle ACB = 90° (Thales' theorem).

This question requires finding circumference given the area. From A = πr² = 196π, we get r² = 196, so r = 14 feet. The circumference is C = 2πr = 2π(14) = 28π feet. A common mistake is taking the square root of 196π instead of just 196. Remember to isolate r² by dividing out π first, then take the square root to find r.

This question asks which point lies on the given circle. The circle has equation (x+1)² + (y-5)² = 9, with center (-1, 5) and radius 3. To check if a point lies on the circle, substitute its coordinates and verify the equation holds. For point (2, 5): (2+1)² + (5-5)² = 3² + 0² = 9 ✓. This confirms (2, 5) lies on the circle. A quick strategy is to check points that are horizontally or vertically aligned with the center, as these calculations are simpler.

This question tests the inscribed angle theorem. An inscribed angle is half the measure of its intercepted arc. Given that the arc measures 110°, the inscribed angle = 110°/2 = 55°. A common error is confusing inscribed angles with central angles - a central angle equals its intercepted arc, but an inscribed angle is always half. Remember: inscribed angle = (1/2) × intercepted arc.

This question asks for the equation of a circle given its center and a point on the circle. First find the radius using the distance formula: r = √[(8-2)² + (3-(-1))²] = √[6² + 4²] = √[36 + 16] = √52. The equation is (x-2)² + (y+1)² = 52. A common error is simplifying √52 unnecessarily or making sign errors with the center coordinates. Keep r² = 52 in the equation rather than approximating.

This question involves a chord and its perpendicular distance from the center. When a perpendicular from the center bisects a chord, it creates a right triangle. The chord AB = 16 units is bisected into two 8-unit segments, and the perpendicular distance is 6 units. Using the Pythagorean theorem: r² = 8² + 6² = 64 + 36 = 100, so r = 10. A key insight is that the perpendicular from the center always bisects the chord. Draw the diagram to visualize the right triangle formed.

This question asks how far a runner travels on a circular track. The runner completes 3/8 of a lap on a track with circumference 400 meters. Distance traveled = (3/8) × 400 = 3 × 50 = 150 meters. A common error is misunderstanding what fraction of a lap means - it's the fraction of the total circumference. When dealing with partial laps, multiply the fraction by the total circumference to find the distance.