Circles Flashcards

A circle with center O has an inscribed polygon ABCD. If arc AB = 80°, arc BC = 100°, and arc CD = 70°, what is the measure of arc DA?

Answer: 110°

All arcs must sum to 360°: arc DA = 360° − 80° − 100° − 70° = 110°.

A line y = mx + b is tangent to the circle x² + y² = r² from the outside. Which condition must hold?

Answer: b² = r²(1 + m²)

The distance from origin to y = mx + b is |b|/√(1+m²) = r, so b² = r²(1+m²).

A circle has area A and circumference C. Which expression is equal to the radius?

Answer: Both A and B

C = 2πr → r = C/(2π); A = πr² → r = √(A/π). Both expressions equal r.

Circle O has diameter AB. Point C lies on the circle. If angle CAB = 35°, what is angle ABC?

Answer: 55°

Angle ACB = 90° (angle in semicircle). So angle ABC = 180° − 90° − 35° = 55°.

A circle with radius 5 is centered at the origin. A point P on the circle is in the first quadrant, and the radius to P makes a 37° angle with the positive x-axis. What are the coordinates of P? (Use sin 37° ≈ 0.6, cos 37° ≈ 0.8)

Answer: Both A and C

P = (5·cos37°, 5·sin37°) = (5·0.8, 5·0.6) = (4, 3). Both A and C are equivalent.

Two concentric circles have radii 3 and 7. A chord of the outer circle is tangent to the inner circle. What is the length of the chord?

Answer: Both A and B

The chord is tangent to the inner circle (radius 3), so its perpendicular distance from center = 3. Half-chord = √(7²−3²) = √40 = 2√10. Full chord = 4√10 = 2√40. Both A and B are equivalent.

The equation of circle K is x² + y² − 4x − 10y + 13 = 0. What is the distance from the center of K to the origin?

Answer: Both B and C

Completing the square: (x−2)² + (y−5)² = 4+25−13 = 16. Center = (2, 5). Distance to origin = √(4+25) = √29. Both B and C are equivalent.