Geometry: Angles, Triangles, Circles, and Coordinates
What this domain tests
Geometry questions cover angle relationships, triangle properties, similarity and congruence, the Pythagorean theorem, area and volume formulas, and coordinate geometry. You should know that the angles of a triangle sum to $180^\circ$, that parallel lines cut by a transversal create equal corresponding and alternate interior angles, and that similar figures have proportional sides and equal angles. Be ready to use the distance and midpoint formulas, the equation of a circle $(x-h)^2+(y-k)^2=r^2$, and slope relationships for parallel (equal slopes) and perpendicular (negative reciprocal slopes) lines. Volume and surface area questions may involve prisms, cylinders, cones, and spheres.
Traps to watch for
A frequent mistake is assuming a triangle is right when it is not, then applying the Pythagorean theorem incorrectly. When figures are similar, side ratios are equal, but areas scale by the square of the ratio and volumes by its cube. Watch for angle pairs that are supplementary rather than equal. In coordinate problems, remember the perpendicular slope is the negative reciprocal, so a slope of $\tfrac23$ pairs with $-\tfrac32$. Do not confuse diameter with radius when using circle formulas.
Worked example
A right triangle has legs $6$ and $8$. Find the hypotenuse and the area. By the Pythagorean theorem, $c^2=6^2+8^2=36+64=100$, so $c=10$. The area is $\tfrac12(6)(8)=24$. Rule of thumb: identify the right angle first, label the hypotenuse as the side opposite it, then apply $a^2+b^2=c^2$ with the hypotenuse alone on one side.