Circles Flashcards

The equation of a circle is (x − 4)² + (y − 4)² = 16. Which of the following points lies on the circle?

Answer: Both A and B

(4−4)² + (8−4)² = 0 + 16 = 16 ✓; (8−4)² + (4−4)² = 16 + 0 = 16 ✓. Both points lie on the circle.

A circle has center (3, −1) and passes through the point (7, 2). What is the area of the circle?

Answer: 25π

Radius = √((7−3)² + (2−(−1))²) = √(16+9) = √25 = 5. Area = π(5²) = 25π.

A chord of a circle with radius 10 is 12 units long. How far is the chord from the center?

Answer: 8

The perpendicular from the center bisects the chord into two segments of 6 each. Distance = √(10² − 6²) = √(100−36) = √64 = 8.

What is the length of the arc intercepted by a central angle of 2π/3 radians in a circle with diameter 18?

Answer: 6π

Radius = 9. Arc length = rθ = 9 × (2π/3) = 6π.

The circle x² + y² + 4x − 12y + 15 = 0 has center (h, k). What is h + k?

Answer: −2 + 6 = 4

Complete the square: (x+2)² − 4 + (y−6)² − 36 + 15 = 0 → (x+2)² + (y−6)² = 25. Center = (−2, 6), so h + k = −2 + 6 = 4.

Two tangent segments are drawn to a circle from an external point P. One tangent touches the circle at A and the other at B. If the arc AB (the minor arc) measures 100°, what is the measure of angle P?

Answer: 80°

The major arc AB = 360° − 100° = 260°. Angle P = (1/2)|major arc − minor arc| = (1/2)(260° − 100°) = (1/2)(160°) = 80°.

A circle has radius 4. A sector of this circle has a perimeter of 4π + 8. What is the central angle of the sector in radians?

Answer: π

Perimeter of sector = arc length + 2 radii = 4θ + 8 = 4π + 8, so 4θ = 4π and θ = π.