A circle has radius 8 cm. A central angle is 90°. What is the arc length? (Use π ≈ 3.14) — 12.56 cm.Arc length = (90°/360°) × 2πr = (1/4) × 2(3.14)(8) = (1/4)(50.24) = 12.56 cm.
A circle has radius 6. The central angle is 60°. What is the sector area? (Leave in terms of π) — 6π.Sector area = (θ/360°) × πr² = (60/360) × 36π = (1/6)(36π) = 6π.
Two inscribed angles intercept the same arc. How do they compare? — They are equal.All inscribed angles intercepting the same arc are equal (all equal half the arc).
A central angle in a circle is 150°. What is the arc length if the radius is 12? Leave in terms of π. — 10π.s = rθ = 12 × (150 × π/180) = 12 × (5π/6) = 10π.
What is the formula for arc length with angle in radians? — s = rθ.Arc length s = rθ, where r is radius and θ is the central angle in radians.
A central angle of 2 radians in a circle of radius 4 subtends an arc of: — 8 units.s = rθ = 4 × 2 = 8 units.
An inscribed angle is 40°. The arc it intercepts is 80°. The remaining arc (reflex arc) is: — 280°.Total circle = 360°. Arc intercepted = 80°. Remaining arc = 360° − 80° = 280°.
An angle inscribed in a circle intercepts an arc of 220°. What is the inscribed angle? — 110°.Inscribed angle = (1/2) × intercepted arc = 220°/2 = 110°.
The circumference of a circle with radius 7 is 14π. What is the arc length for a 90° central angle? — Both a and b are equivalent.Arc = (90/360) × 14π = (1/4)(14π) = 3.5π = 7π/2. Both a and b are equivalent expressions.
Convert x² + y² − 4x + 6y − 12 = 0 to standard form. What is the center? — (2, −3).Complete the square: (x²−4x+4) + (y²+6y+9) = 12+4+9 = 25. (x−2)² + (y+3)² = 25. Center: (2, −3).
From x² + y² − 4x + 6y − 12 = 0, what is the radius? — Both a and c.(x−2)² + (y+3)² = 25. r² = 25 → r = 5. Both 5 and √25 = 5 are correct.
What is the first step in converting x² + y² + 8x − 2y − 8 = 0 to standard form? — Group x-terms and y-terms, then complete the square.Group: (x² + 8x) + (y² − 2y) = 8. Then complete the square for each group.
Complete the square for x² − 10x. The completed square expression is: — (x − 5)² − 25.Take half of −10, square it: (−5)² = 25. x² − 10x + 25 = (x − 5)². But we must subtract 25 to keep equality: (x − 5)² − 25.
Complete the square for y² + 6y to find what to add to both sides. — 9.Half of 6 is 3; 3² = 9. Add 9 to both sides to complete y² + 6y + 9 = (y + 3)².
From the previous equation, what is the radius? — √9 = 3.r² = 9, so r = 3. Both option a (3) and option d (√9 = 3) are the same.
Which point is NOT on the circle (x − 2)² + y² = 9? — (0, 0).Check (0,0): (0−2)² + 0² = 4 ≠ 9. (5,0): (3)²=9✓. (2,3): 0+9=9✓. (2,−3): 0+9=9✓. So (0,0) is NOT on the circle.
What does completing the square allow you to do? — Convert general form to standard form to identify center and radius.Completing the square converts x² + Dx → (x + D/2)² − (D/2)², revealing the standard form and thus the center and radius.
A circle with equation (x + 3)² + (y − 1)² = r² passes through (0, 1). What is r? — Both a and c.Substitute (0, 1): (0+3)² + (1−1)² = 9. r² = 9, r = 3.
A quadrilateral has vertices A(0,0), B(4,0), C(4,3), D(0,3). To prove it is a rectangle: — Both b and c are valid methods.A rectangle can be proven by showing perpendicular adjacent sides OR by showing it's a parallelogram with equal diagonals.
Slope of the line through (−2, 3) and (4, 9) is: — 1.Slope = (9−3)/(4−(−2)) = 6/6 = 1.
A triangle has vertices at (0,0), (6,0), and (3,4). What formula proves it is isosceles? — Distance formula (show two sides are equal).To prove isosceles: use the distance formula to show two sides have equal length.
Vertices of a quadrilateral: A(0,0), B(5,0), C(6,3), D(1,3). To prove it is a parallelogram: — Both b and c work.A parallelogram can be proven by: (1) opposite sides are parallel (equal slopes), OR (2) diagonals bisect each other (same midpoint).
The distance from (a, b) to the origin is: — √(a² + b²).d = √((a−0)² + (b−0)²) = √(a² + b²).
Vertices A(1,1), B(5,1), C(5,4), D(1,4) form what shape? — Rectangle.AB is horizontal, BC is vertical — adjacent sides are perpendicular (slopes: 0 and undefined). This is a rectangle.
What is the slope of the line perpendicular to y = 3x − 2? — −1/3.Slope of y = 3x − 2 is 3. Perpendicular slope = −1/3 (negative reciprocal of 3).
The diagonals of a square are: — Equal in length and perpendicular bisectors of each other.A square's diagonals are equal in length, perpendicular to each other, and bisect each other.
To prove a quadrilateral is NOT a parallelogram, you could show: — Opposite sides have different slopes (not parallel).If opposite sides don't have equal slopes, they aren't parallel, proving the quadrilateral is NOT a parallelogram.