SAT® · Test Mechanics: The Tools and the Traps
What Is Not on the Reference Sheet
The reference sheet on the math test skips slope, distance, midpoint, and the quadratic formula. You're expected to rebuild them from a right triangle, from completing the square, and from memory.
The formula sheet printed on the test gives you area and volume formulas, the Pythagorean theorem, and a few circle and triangle facts. It does not give you slope, distance, midpoint, or the quadratic formula. That's not an oversight. Those tools all come from one idea you're expected to already own: a right triangle hiding between two points. Drop a horizontal leg and a vertical leg between any two points, and the Pythagorean theorem hands you the distance as the hypotenuse. Average the two legs' endpoints and you get the midpoint. Divide the vertical leg by the horizontal leg and you get slope. One triangle, three formulas, zero need for a sheet.
Circles work the same way: the sheet gives you the Pythagorean theorem, which is exactly what you need to complete the square. An equation like x2 + y2 - 6x + 4y = 12 isn't in a recognizable circle shape yet. Group the x terms and y terms, add the square of half each coefficient to both sides, and the left side becomes (x-3)2 + (y+2)2, matching center-radius form. Here that gives (x-3)2 + (y+2)2 = 25, so the center is (3, -2) and the radius is 5. The sheet trusts you to do this algebra instead of handing you a shortcut.
The quadratic formula gets the same treatment: it's missing, so when factoring fails you either complete the square on the spot or have the formula memorized cold. The same logic extends to circles cut into pieces. A full circle is 360 degrees, so any arc length or sector area is just that fraction of the circumference or area formulas the sheet does give you.
Key terms
- Center-radius form
- A circle equation written as (x-h)2 + (y-k)2 = r2, showing center (h,k) and radius r directly.
- Completing the square
- Adding the square of half the linear coefficient to turn x2 + bx into a perfect square trinomial.
- Midpoint
- The point exactly halfway between two coordinates, found by averaging their x-values and averaging their y-values.
- Pythagorean theorem
- In a right triangle, a2 + b2 = c2, where c is the hypotenuse; it generates the distance formula.
Watch it explained
- How it worksHow do slope, distance and midpoint all fall out of one right triangle drawn between two points?
- How it worksHow does completing the square turn x2 + y2 - 6x + 4y = 12 into a circle with a center and a radius?
- Worked exampleFind the distance and the midpoint between (-2, 5) and (6, -1) with no formula sheet to lean on.
- Common mistakeWhy isn't the quadratic formula printed on the SAT reference sheet, and what do you do about that?
- Going furtherWhat lets you rebuild arc length and sector area from the fact that a circle holds 360 degrees?
Check yourself
End of topic.