The question asks for the measure of angle ACB in triangle ABC, where AB is the diameter and C is on the circle. The theorem states that an angle inscribed in a semicircle is a right angle, so angle at C is 90°. This is because the central angle is 180° for the diameter. The diagram confirms the setup. Common errors include assuming isosceles properties leading to 45° or 60°, or thinking it's 180° at the center. Recall the inscribed angle is half the central angle. When seeing diameter and point on circle, immediately recognize the 90° angle.

The question asks for the equation of a circle with center (-4, 2) and radius 3. The standard form is (x - h)² + (y - k)² = r², so (x + 4)² + (y - 2)² = 9. Note the signs for the center. Common errors include wrong signs, like (x - 4)² + (y + 2)², or using r instead of r², getting =3. Another is mixing coordinates. Always plug in the center to verify: at x = -4, y = 2, it should equal r². Distinguish radius from diameter; here r = 3, not 6.

The question asks for the central angle of a sector with radius 5 in and area 25π/4 in². The sector area formula is (θ/360) × πr²; solve for θ. With r = 5, full area π × 25 = 25π, so (θ/360) × 25π = 25π/4, θ/360 = 1/4, θ = 90°. The fraction simplifies directly. A common error is using arc length or forgetting to multiply by 360, leading to 45° or 180°. Another is confusing r with diameter, doubling area. Emphasize isolating θ correctly and using radius in r². Rearrange the formula before substituting to avoid errors.

The question asks for the radius of a circle with chord AB = 12 and perpendicular distance from center O to AB of 8. Use the property that the perpendicular bisects the chord, so apply Pythagoras: r² = (chord/2)² + d², where d = 8. Half chord is 6, so r² = 36 + 64 = 100, r = 10. This forms a right triangle with legs 6 and 8. Common errors include not halving the chord, yielding r² = 144 + 64 = 208, r = √208. Another is subtracting, getting 6. Always halve the chord and use radius in the hypotenuse. Visualize the triangle to ensure correct application.

The question asks for the area of a circle centered at the origin passing through (0, -6). The radius is the distance from center to the point: √(0² + (-6)²) = 6, so r = 6. The area formula is πr², yielding π × 36 = 36π square units. The point (0, -6) is on the y-axis, confirming r = 6. Common errors include using diameter 12, leading to area π × 144 = 144π or circumference 12π. Another is squaring incorrectly, like π × 6 = 6π. Distinguish radius as the distance, not confusing with coordinates. Plot the point to visualize the radius clearly.

The question asks for the measure of the angle between radius OT and tangent line ℓ at point T. The key property is that a tangent to a circle is perpendicular to the radius at the point of tangency, so the angle is 90°. This is a fundamental theorem in circle geometry. The diagram shows OT drawn to T, confirming the right angle. Common errors include assuming it's 45° or 60° from triangle misconceptions, or 180° thinking it's straight. Always recall that tangent-radius perpendicularity distinguishes it from secants. When seeing tangents, immediately think 90° for such angles to avoid confusion.

The question asks for the equation of a circle centered at (2, -1) passing through (8, 3). The standard form is (x - h)² + (y - k)² = r²; first find r by distance from center to point. Distance: √[(8-2)² + (3 - (-1))²] = √(36 + 16) = √52, so r² = 52. Thus, (x - 2)² + (y + 1)² = 52. Common errors include sign mistakes in the center, like (x + 2)² + (y - 1)², or using r instead of r², getting (√52)² = 52 but halving to 26. Another is miscalculating distance, forgetting to square. Verify by plugging the point back into the equation to confirm.

The question asks for the perpendicular distance from the center O to chord AB in circle O, where AB = 16 and radius = 10. The property used is that the perpendicular from the center to a chord bisects the chord, and Pythagoras theorem applies: d² + (chord/2)² = r², where d is the distance and r is the radius. Half the chord is 8, so d² + 8² = 10², d² + 64 = 100, d² = 36, d = 6. This distance is along the radius to the midpoint, perpendicular to AB. Common errors include not halving the chord, leading to d² + 16² = 10² (impossible), or confusing with radius, getting 10 or 8. Always remember to distinguish radius from other lengths and halve the chord in the theorem. Sketching the right triangle helps visualize and avoid mistakes.

The question asks for the length of the border around a circular garden with a diameter of 18 ft, which is the circumference. The circumference can be calculated using C = πd or C = 2πr, where d is the diameter and r is the radius; here, emphasize using the diameter directly for simplicity since it's given. With d = 18 ft, C = π × 18 = 18π ft; alternatively, r = 9 ft, so C = 2π × 9 = 18π ft. The border is the perimeter, matching the circumference. A key error is confusing circumference with area, leading to πr² = 81π, or doubling incorrectly to 36π. Another mistake is using half the diameter as radius incorrectly. When the diameter is provided, opt for C = πd to avoid radius calculation errors.

The question asks for the center and radius of the circle given by the equation (x + 3)² + (y - 4)² = 25. The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. To identify, rewrite the equation to match the standard form, revealing h = -3, k = 4, and r² = 25, so r = 5. Thus, the center is (-3, 4) and the radius is 5. Common errors include sign mistakes, such as interpreting as center (3, -4), or confusing r with r², leading to r = 25. Always double-check the signs in the equation and remember that the right side is r², not r. A useful strategy is to expand the standard form to verify if needed.