A central angle has its vertex at: — The center of the circle.A central angle has its vertex at the center of the circle. Its measure equals the arc it intercepts.
An inscribed angle has its vertex: — On the circle itself.An inscribed angle has its vertex on the circle, with both sides as chords of the circle.
An inscribed angle equals what fraction of its intercepted arc? — Half the arc.Inscribed Angle Theorem: inscribed angle = (1/2) × intercepted arc.
A central angle intercepts an arc of 100°. What is the measure of an inscribed angle intercepting the same arc? — 50°.Inscribed angle = arc/2 = 100°/2 = 50°.
An inscribed angle measures 35°. What is the measure of the intercepted arc? — 70°.Arc = 2 × inscribed angle = 2 × 35° = 70°.
An inscribed angle in a semicircle always measures: — 90°.An inscribed angle intercepting a semicircle (arc = 180°) equals 180°/2 = 90°. It's always a right angle!
The diameter of a circle subtends an inscribed angle of exactly: — 90°.The diameter corresponds to an arc of 180°. Inscribed angle = 180°/2 = 90°. (Thales' Theorem)
What is the area of a full circle with radius r, written in the sector formula? — Both a and b are equivalent.A full circle = sector with θ = 360°. Area = (360/360) × πr² = πr². Both forms are equivalent.
In a circle, a chord that passes through the center is called: — A diameter.A chord passing through the center is the diameter — the longest chord in a circle.
Two radii form a central angle of 45°. What fraction of the circle does the sector represent? — 1/8.45°/360° = 1/8. The sector is 1/8 of the full circle.
What is the relationship between a central angle and its intercepted arc? — Central angle = arc.A central angle is EQUAL to its intercepted arc (both measured in degrees). This is the definition.
The standard equation of a circle with center (h, k) and radius r is: — (x − h)² + (y − k)² = r².Standard form: (x − h)² + (y − k)² = r². Note the MINUS signs — (x + 1)² means the center is at x = −1.
The equation x² + y² = 25 represents a circle with: — Center (0, 0), r = 5.x² + y² = 25 has center at the origin (0, 0) and r = √25 = 5.
What is the center of (x − 3)² + (y + 2)² = 16? — (3, −2).Compare to (x − h)² + (y − k)²: h = 3, k = −2. Center is (3, −2). Note: (y + 2) = (y − (−2)).
What is the radius of (x + 1)² + (y − 5)² = 49? — Both b and c.r² = 49, so r = √49 = 7. Both b and c are the same answer.
Write the equation of a circle centered at (2, −3) with radius 4. — (x − 2)² + (y + 3)² = 16.Center (2, −3), r = 4, r² = 16. Equation: (x − 2)² + (y − (−3))² = 16 → (x − 2)² + (y + 3)² = 16.
Does the point (3, 4) lie on the circle x² + y² = 25? — Yes.3² + 4² = 9 + 16 = 25 ✓. The point (3, 4) satisfies the equation, so it lies on the circle.
The equation of a circle with center at the origin and radius 11 is: — x² + y² = 121.Center (0, 0), r = 11, r² = 121. Equation: x² + y² = 121.
Two circles have the same radius but different centers. They are: — Congruent.All circles with the same radius are congruent — they have the same shape and size, just different positions.
In coordinate geometry proofs, which formula proves two segments are equal in length? — Distance formula.The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) is used to prove that two segments have equal length.
To prove two lines are parallel in the coordinate plane: — Show they have equal slopes.Parallel lines have equal (identical) slopes. Non-vertical parallel lines always have the same slope.
To prove two lines are perpendicular in the coordinate plane: — Show their slopes are negative reciprocals (m₁ × m₂ = −1).Two lines are perpendicular if and only if their slopes are negative reciprocals: m₁ × m₂ = −1.
To prove two segments bisect each other: — Show the midpoints of both segments are the same point.Two segments bisect each other if they share the same midpoint. Use the midpoint formula on each segment.
The midpoint of segment joining (2, 6) and (8, 4) is: — (5, 5).M = ((2+8)/2, (6+4)/2) = (10/2, 10/2) = (5, 5).
The distance between (1, 3) and (5, 6) is: — Both a and c.d = √((5−1)² + (6−3)²) = √(16+9) = √25 = 5.
The slope of the line from (0, 0) to (a, 0) where a ≠ 0 is: — 0.Slope = (0−0)/(a−0) = 0. Any horizontal line has slope = 0.
Slope of the line y = −4 is: — 0.y = −4 is a horizontal line. All horizontal lines have slope = 0.
Slope of the line x = 5 is: — Undefined.x = 5 is a vertical line. Vertical lines have undefined slope (you can't divide by Δx = 0).