A circle's equation shows its center with flipped signs and its radius squared

Rewrite a circle's equation in center-radius form

A circle has equation $(x-h)^2+(y-k)^2=r^2$, with center $(h,k)$ and radius $r$. For $x^2+y^2-4x+6y=12$, complete the squares to get $(x-2)^2+(y+3)^2=25$.

The center is $(2,-3)$ and the radius is $5$. In SAT Desmos, graph the original equation and the center point to check the position. The top of the circle is $(2,2)$, five units above the center.

$\displaystyle \sqrt{12+4+9}$

The radius after completing both squares.

$\displaystyle \left(2,-3\right)$

The center, not a point on this circle.

Use coordinate differences for lengths and slopes

Between $(1,-2)$ and $(7,6)$, the distance is $\sqrt{(7-1)^2+(6+2)^2}=10$ and the slope is $(6+2)/(7-1)=4/3$. A vertical line has an undefined slope because its horizontal change is zero.

Parallel nonvertical lines have the same slope. Perpendicular nonvertical lines with nonzero slopes have slopes whose product is $-1$. Horizontal and vertical lines form the special perpendicular pair.

Unequal axis scales distort shapes, so rely on coordinates for exact answers

An unequal horizontal and vertical scale can make a circle look like an ellipse. Use the equation and numerical coordinates for exact conclusions; a drawing's apparent angle or length is not proof.

For areas and volumes, identify the correct formula and units before calculating. The SAT reference sheet supplies common geometry formulas, but you still need to distinguish a radius from a diameter, a slant height from a perpendicular height, and square units from cubic units.

Keep practicing the connected skills