Circles Flashcards

Write the equation of a circle with center (7, −3) and radius 4.

Answer: (x − 7)² + (y + 3)² = 16

Standard form: (x−7)²+(y−(−3))²=4² → (x−7)²+(y+3)²=16.

A circle has a central angle of 240° and a radius of 3. What is the arc length?

Answer: 4π

Arc = (240/360)(2π·3) = (2/3)(6π) = 4π.

An inscribed angle of 65° intercepts an arc. The central angle subtending the same arc is:

Answer: 130°

Central angle = 2 × inscribed angle = 2 × 65° = 130°.

The diameter of a circle is the chord passing through the center. If an inscribed angle intercepts the diameter, what is the angle's measure?

Answer: 90°

The diameter subtends a semicircle (180°), and an inscribed angle is half the arc = 90° (Thales' theorem).

Rewrite x² + y² + 12x − 16y + 75 = 0 in standard form. What is the radius?

Answer: 5

(x+6)²+(y−8)²=36+64−75=25; radius=√25=5.

Two tangent lines from an external point each touch a circle of radius 5. The distance from the external point to the center is 13. What is the length of each tangent segment?

Answer: 12

Tangent is perpendicular to radius at the point of tangency; length = √(13²−5²) = √(169−25) = √144 = 12.

A circle has equation (x−2)²+(y+6)²=r². The point (6, −6) lies on the circle. What is r?

Answer: 4

(6−2)²+(−6+6)²=16+0=16, so r²=16, r=4.