To find the equation of a circle with center (2,5) passing through (8,1), we first need the radius. The radius is the distance from center to the given point: r = √[(8-2)² + (1-5)²] = √[6² + (-4)²] = √[36 + 16] = √52. The standard form equation is (x-h)² + (y-k)² = r², so with center (2,5) and r² = 52, the equation is (x-2)² + (y-5)² = 52. Remember to keep r² = 52 rather than trying to simplify √52, as the equation uses r².

We need to find the arc length (crust) of a pizza slice with radius 8 inches and central angle 45°. The arc length formula is s = (θ/360°) × 2πr, where θ is in degrees. Substituting: s = (45°/360°) × 2π(8) = (1/8) × 16π = 2π inches. The key is recognizing that 45° is 1/8 of a full circle (360°), so the arc is 1/8 of the circumference. When solving arc length problems, always express the angle as a fraction of 360° first.

This problem involves a triangle inscribed in a circle where one side is the diameter. When a triangle is inscribed in a circle with one side as the diameter, the angle opposite the diameter is always 90° (Thales' theorem). Given AB = 20 (diameter), AC = 12, and BC = 16, triangle ABC is a right triangle with the right angle at C. The area = (1/2) × base × height = (1/2) × 12 × 16 = 96. Note that 12² + 16² = 144 + 256 = 400 = 20², confirming the right triangle. Always check if a triangle inscribed in a semicircle forms a right triangle.

We need to find the equation of a circle centered at the origin that passes through (3,-4). For a circle centered at the origin, the equation is x² + y² = r². Since the circle passes through (3,-4), we can find r by calculating the distance from the origin: r = √(3² + (-4)²) = √(9 + 16) = √25 = 5. Therefore, r² = 25, and the equation is x² + y² = 25. A common error is forgetting to square the radius in the final equation or making calculation errors when finding the distance.

This question asks for the measure of an inscribed angle given the arc it intercepts. The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. Since arc AB measures 110°, the inscribed angle ∠ACB = 110°/2 = 55°. This fundamental theorem is often confused with central angles, which equal their intercepted arcs. Remember: inscribed angles are always half their intercepted arc, regardless of where on the circle the angle's vertex is located.

This problem involves a tangent line to a circle, where we need to find the distance from the center O to a point Q on the tangent. Since a tangent is perpendicular to the radius at the point of tangency, triangle OPQ is a right triangle with the right angle at P. Given OP = 10 (radius) and PQ = 24, we use the Pythagorean theorem: OQ² = OP² + PQ² = 10² + 24² = 100 + 576 = 676. Therefore, OQ = √676 = 26. The key property is that tangent lines are always perpendicular to radii at the point of tangency.

This question asks us to identify the center and radius from the standard form equation of a circle. The standard form is (x-h)² + (y-k)² = r², where (h,k) is the center and r is the radius. From (x-4)² + (y+1)² = 49, we can identify: h = 4, k = -1 (note that y+1 = y-(-1)), and r² = 49, so r = 7. The center is (4,-1) and the radius is 7. A common mistake is confusing the signs in the center coordinates or thinking r² is the radius instead of taking the square root.

The problem asks for the area of a circle given its diameter of 18 inches. The area formula for a circle is A = πr², where r is the radius. Since diameter = 2r, we have r = 18/2 = 9 inches. Substituting into the formula: A = π(9)² = 81π square inches. The most common error is using the diameter directly in the area formula instead of first finding the radius. When given diameter, always divide by 2 to find radius before calculating area.

This problem asks for the length of a chord when we know the radius and the distance from the center to the chord. When a radius is perpendicular to a chord, it bisects the chord, creating two congruent right triangles. Using the right triangle formed by the radius (13 cm), the distance from center to chord (5 cm), and half the chord length, we apply the Pythagorean theorem: 13² = 5² + (BC/2)². Solving: 169 = 25 + (BC/2)², so (BC/2)² = 144, giving BC/2 = 12, and therefore BC = 24 cm. A common error is using the distance to the chord as the full chord length instead of recognizing the perpendicular bisector relationship.

This problem involves the intersecting chords theorem, which states that when two chords intersect inside a circle, the products of their segments are equal: AE × EB = CE × ED. Given AE = 3, EB = 12, and CE = 4, we can solve for ED: 3 × 12 = 4 × ED, so 36 = 4 × ED, giving ED = 9. This theorem is a powerful tool for finding unknown segments when chords intersect. Always identify which segments belong to which chord before applying the formula.