In a cyclic quadrilateral (inscribed in a circle), opposite angles are: — Supplementary (sum to 180°).In a cyclic quadrilateral, opposite angles are supplementary — they sum to 180°. This is a key theorem.
A chord of a circle is 90° from the center. The inscribed angle intercepting the same chord as a diameter is: — 90°.Any inscribed angle in a semicircle (intercepting a diameter) = 90°. This is Thales' theorem.
A circle has radius 5 cm and sector area 10π cm². What is the central angle in radians? — 2π/5.A = (1/2)r²θ → 10π = (1/2)(25)θ → θ = 20π/25 = 4π/5. Wait: (1/2)(25)(θ) = 10π → 12.5θ = 10π → θ = 10π/12.5 = 4π/5. So angle = 4π/5 radians.
A sector has area 36π and radius 6. What is the central angle? — 2π radians.A = (1/2)r²θ → 36π = (1/2)(36)θ = 18θ → θ = 2π radians (= 360°). This is the full circle!
A circle has equation (x − 4)² + (y + 7)² = 81. Does the point (4, 2) lie inside, on, or outside? — Inside (distance < r).Distance from center (4, −7) to (4, 2): √((4−4)² + (2−(−7))²) = √81 = 9 = r. Point is ON the circle!
The equation (x − 1)² + (y − 2)² = −5 describes: — No real circle exists.r² must be positive. r² = −5 < 0 is impossible. This equation has no real graph.
If r² = 0 in the standard equation, the graph is: — A point (degenerate circle).r² = 0 means r = 0 — a circle of zero radius, which is just a single point (the center). This is a degenerate circle.
Find the radius of x² + y² − 6x − 8y = 0. — r = 5.Complete the square: (x−3)² + (y−4)² = 9+16 = 25. r = 5.
A circle passes through (0, 0) and has center (3, 4). What is the equation? — (x−3)² + (y−4)² = 25.r = distance from center (3,4) to (0,0) = √(9+16) = 5. r² = 25. Equation: (x−3)² + (y−4)² = 25.
The general form of a circle equation is x² + y² + Dx + Ey + F = 0. The center is: — (−D/2, −E/2).Completing the square: center = (−D/2, −E/2). The negative signs come from the standard form (x−h)².
A strategic choice for placing a triangle in a coordinate proof is: — One vertex at the origin, one side along the x-axis.Placing one vertex at the origin and a side along the x-axis gives simpler coordinates (like (0,0), (a,0), (b,c)) making algebra cleaner.
In a general coordinate proof, vertices use variables like (a, 0) and (0, b) to: — Keep the proof general for ALL such triangles.General coordinates make the proof valid for ALL triangles of that type, not just one specific example.
The midpoint of the hypotenuse of a right triangle with vertices (0,0), (2a,0), (0,2b) is: — (a, b).Hypotenuse goes from (2a, 0) to (0, 2b). Midpoint = ((2a+0)/2, (0+2b)/2) = (a, b).
To prove a quadrilateral is a RHOMBUS using coordinates: — Show all four sides have equal length.A rhombus has all four sides equal. Use the distance formula to show all four side lengths are equal.
To prove a triangle is a RIGHT triangle using coordinates: — Show two sides have perpendicular slopes (slopes are negative reciprocals).Prove two sides are perpendicular using slopes: m₁ × m₂ = −1. Those sides form the right angle.
The centroid (average of vertices) of a triangle with vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃) is: — ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).The centroid is the average of all three vertices: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).