SAT: Circles in Geometry & Coordinates

Understand the idea

Parent circle: $\displaystyle x^{2} + y^{2} = 1$. Translating its center and scaling its radius gives $\displaystyle \left(x - h\right)^{2} + \left(y - k\right)^{2} = r^{2}$. The center is (h, k) and $\displaystyle r > 0$ is the radius.

Choose and justify a method

For $\displaystyle x^{2}$ + Dx, add $\displaystyle \left(\frac{D}{2}\right)^{2}$ to form $\displaystyle \left(x + \frac{D}{2}\right)^{2}$. Add the same quantity on the other side; do this separately for x and y. The right side after completing squares equals $\displaystyle r^{2}$, not r.

Check the conditions

Substitute a point into the left side to compute squared distance from the center. Equality with $\displaystyle r^{2}$ means on the circle; smaller means inside; larger means outside. A negative squared radius gives no real circle.

Common pitfalls

Explain it to yourself

Explain why your method works in circle equations & completing the square. State one condition it needs and describe what can go wrong without that condition.

Understand the idea

A central angle has its vertex at the center. An inscribed angle has its vertex on the circle and measures half its intercepted arc (the arc not containing its vertex). Equal intercepted arcs give equal inscribed angles.

Choose and justify a method

A tangent meets the circle at one point and is perpendicular to the radius there. A radius perpendicular to a chord bisects the chord. Draw that right triangle before using the Pythagorean theorem.

Check the conditions

For intersecting chords inside a circle, products of segment lengths agree: $\displaystyle PA \times PB = PC \times PD$. Tangent segments from the same external point are equal. Check that the diagram meets a theorem’s hypotheses before applying it.

Common pitfalls

Explain it to yourself

Explain why your method works in circle angles, chords & tangents. State one condition it needs and describe what can go wrong without that condition.

Understand the idea

One full turn is $\displaystyle 360^{\circ}$ or $\displaystyle 2\, \pi$ radians. Multiply degrees by $\displaystyle \frac{\pi}{180}$ to convert to radians. A radian measures arc length divided by radius.

Choose and justify a method

With $\displaystyle \theta$ in radians, arc length $\displaystyle s = r\, \theta$ and sector area $\displaystyle A = \frac{r^{2}\, \theta}{2}$. With degrees, use the fraction $\displaystyle \frac{\theta}{360}$ of the circumference or area. Do not put a degree value directly into the radian formula.

Check the conditions

On the unit circle, the point at angle $\displaystyle \theta$ is $\displaystyle \left(\operatorname{cos}\, \theta, \operatorname{sin}\, \theta\right)$. The signs come from its quadrant. Scaling to radius r multiplies both coordinates by r; translating adds the center coordinates.

Common pitfalls

Explain it to yourself

Explain why your method works in arcs, sectors & radians. State one condition it needs and describe what can go wrong without that condition.