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Welcome back to circles! Today we'll use the equation of a circle in the coordinate plane and work with chords, tangents, central and inscribed angles, and arc length.
A circle is the set of all points in a plane that lie a fixed distance r, the radius, from a fixed point (h, k), the center. Writing that definition with the distance formula gives the equation of a circle: (x - h)² + (y - k)² = r².
A chord is a segment joining two points on the circle, and the diameter is the longest chord because it passes through the center. A tangent line touches the circle at exactly one point and is always perpendicular to the radius drawn to that point.
Angles come next. A central angle has the same measure as the arc it intercepts, while an inscribed angle that intercepts the same arc measures exactly half as much. Arc length is that fraction of the whole circumference: arc = (central angle / 360°) × 2πr.
A circle has center (3, -2) and passes through the point (7, 1). Write the equation of the circle, then find the length of the arc intercepted by a central angle of 90°.
- Find the radius with the distance formula: r = √((7 - 3)² + (1 - (-2))²) = √(16 + 9) = 5.
- Write the equation using center (h, k) = (3, -2) and r = 5: (x - 3)² + (y + 2)² = 25.
- Recall that arc length is a fraction of the circumference: arc = (central angle / 360°) × 2πr.
- Substitute the values: arc = (90/360) × 2π(5) = (1/4)(10π) = 2.5π.
- Write the final answers: the circle is (x - 3)² + (y + 2)² = 25, and the arc length is 2.5π ≈ 7.85 units.
