Coordinate geometry and circle equations: use the distance and midpoint formulas and write or interpret the standard-form equation of a circle, completing the square when needed to find the center and radius.

What this skill is asking

Coordinate geometry brings algebra and geometry together in the $xy$-plane. The Digital SAT (Geometry and Trigonometry domain) tests the distance and midpoint formulas and the standard-form equation of a circle, including using completing the square to recover a circle's center and radius from a general equation. These connect directly to the Pythagorean theorem and to the equivalent-expressions skill.

The coordinate formulas

Three formulas, all rooted in the Pythagorean theorem.

A worked completing-the-square

Turning a general circle equation into standard form exposes the center and radius.

Distance, midpoint, and the Pythagorean link

The distance formula is just the Pythagorean theorem in disguise: the horizontal gap $(x_2 - x_1)$ and the vertical gap $(y_2 - y_1)$ are the legs of a right triangle whose hypotenuse is the distance. That is why the $3$-$4$-$5$ and $5$-$12$-$13$ triples show up in distance answers. The midpoint is the average of the endpoints' coordinates, which the SAT uses for "the center of a segment" or, since a circle's center is the midpoint of any diameter, to find a circle's center from the endpoints of a diameter. Recognising distance as Pythagoras and midpoint as averaging keeps both formulas easy to reconstruct.

Reading and building circle equations

The standard form $(x - h)^2 + (y - k)^2 = r^2$ is built so the center and radius are visible: the center is $(h, k)$ (note the sign flip, so $(x + 3)$ means $h = -3$), and the radius is the square root of the constant on the right (so $= 16$ means $r = 4$, not $16$). To build a circle's equation, you need its center and radius; if given the center and a point on the circle, the radius is the distance between them. When a circle is presented expanded (as $x^2 + y^2 + Dx + Ey + F = 0$), completing the square in $x$ and $y$ returns it to standard form so you can read the center and radius, the single most common circle-equation task on the test.

Points on a circle and the diameter

Two more coordinate-circle ideas appear regularly. First, a point lies on a circle exactly when its coordinates satisfy the equation, so to test whether $(7, 2)$ is on a circle, substitute and check that the two sides are equal; to find where a circle meets an axis, set $x = 0$ or $y = 0$ and solve. Second, because a circle's center is the midpoint of any diameter, you can find the center from the two endpoints of a diameter by averaging their coordinates, and the radius is half the diameter's length (use the distance formula on the endpoints, then halve). So a question that gives the endpoints of a diameter is really a midpoint-and-distance question feeding into the standard-form equation. Linking the circle equation to the distance and midpoint formulas this way means a single small toolkit, distance, midpoint, and completing the square, covers every coordinate-circle question the SAT poses.