Circles

Four core circle facts:

1. Circumference = $2\pi r$ = $\pi d$ (where $d$ is diameter).

2. Area = $\pi r^2$.

3. Equation of a circle centered at $(h, k)$ with radius $r$:

$(x - h)^2 + (y - k)^2 = r^2$

The SAT often gives you a non-standard form like $x^2 + y^2 - 4x + 6y = 12$ and asks you to find the center or radius. Complete the square for both $x$ and $y$ to convert to standard form.

4. Central angles, arcs, and sectors.

• A central angle has its vertex at the center.
• The arc length subtended by a central angle of $\theta$ degrees is $\frac{\theta}{360} \cdot 2\pi r$.
• The sector area (pie slice) is $\frac{\theta}{360} \cdot \pi r^2$.

In radians: arc length = $r\theta$ and sector area = $\frac{1}{2} r^2 \theta$.

Key relationship: the fraction of the circle is the same for all three measures (angle, arc, sector area). If a central angle is one-fourth of the full circle (90°), then the arc is one-fourth of the circumference, and the sector is one-fourth of the area.

Inscribed angle theorem (sometimes tested):

• An inscribed angle has its vertex on the circle.
• An inscribed angle is half the central angle subtending the same arc.

Example: if a central angle is 80°, an inscribed angle facing the same arc is 40°.

Tangent line to a circle: perpendicular to the radius at the point of tangency. If a tangent touches the circle at point $P$, then the radius to $P$ meets the tangent at 90°.

Completing the square refresher: to convert $x^2 - 4x$ to a perfect square, take half of the linear coefficient (-4 / 2 = -2), square it (4), and add: $x^2 - 4x + 4 = (x - 2)^2$.

For $x^2 + y^2 - 4x + 6y = 12$:

• Group: $(x^2 - 4x) + (y^2 + 6y) = 12$.
• Complete: $(x^2 - 4x + 4) + (y^2 + 6y + 9) = 12 + 4 + 9$.
• Rewrite: $(x - 2)^2 + (y + 3)^2 = 25$.
• Center $(2, -3)$, radius 5.