SAT Math formula sheet · all 24 · Geometry and Trigonometry · #24 of 24

Circles on the SAT: How It Is Tested

Circles

Circles diagram for the SAT: The right side of (x - h)^2 + (y - k)^2 = r^2 is the radius SQUARED. A radius of 3m makes the right side 9m^2, not 3m^2,
What it is really testingThe right side of (x - h)^2 + (y - k)^2 = r^2 is the radius SQUARED. A radius of 3m makes the right side 9m^2, not 3m^2, and a right side of 49k^2 means the radius is 7k. The center goes in with the signs flipped: center (5, 12) is (x - 5) and (y - 12). Two traps in one question, and both wrong answers are sitting in the choices.

The question from the sheet

Hard
SAT Geometry and Trigonometryanswer B
A circle in the xyxy-plane has its center at (5,12)(5, 12) and has a radius of 3m3m. Which of the following is the equation of this circle?
A(x+5)2+(y+12)2=9m2(x + 5)^{2} + (y + 12)^{2} = 9m^{2}
✓(x−5)2+(y−12)2=9m2(x - 5)^{2} + (y - 12)^{2} = 9m^{2}
C(x−5)2+(y−12)2=3m2(x - 5)^{2} + (y - 12)^{2} = 3m^{2}
D(x+5)2+(y+12)2=3m2(x + 5)^{2} + (y + 12)^{2} = 3m^{2}

How the diagram applies

  1. Center (5,12)(5, 12) goes in with flipped signs: (x−5)2+(y−12)2(x - 5)^{2} + (y - 12)^{2}. That rules out the two choices with (x+5)(x + 5) and (y+12)(y + 12).
  2. The radius is 3m3m, and the equation wants r2r^{2}: (3m)2=9m2(3m)^{2} = 9m^{2}, not 3m23m^{2}.
  3. So the circle is (x−5)2+(y−12)2=9m2(x - 5)^{2} + (y - 12)^{2} = 9m^{2}. Same idea in reverse: if a test gives you =49k2= 49k^{2}, the radius is 7k7k, not 49k49k.

Answer: B

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Common Questions

It comes up as an SAT circles question. The right side of (x - h)^2 + (y - k)^2 = r^2 is the radius SQUARED. A radius of 3m makes the right side 9m^2, not 3m^2, and a right side of 49k^2 means the radius is 7k. The center goes in with the signs flipped: center (5, 12) is (x - 5) and (y - 12). Two traps in one question, and both wrong answers are sitting in the choices.

Center (5, 12) goes in with flipped signs: (x - 5)^2 + (y - 12)^2. That rules out the two choices with (x + 5) and (y + 12). The radius is 3m, and the equation wants r^2: (3m)^2 = 9m^2, not 3m^2. So the circle is (x - 5)^2 + (y - 12)^2 = 9m^2. Same idea in reverse: if a test gives you = 49k^2, the radius is 7k, not 49k.

No. The SAT reference sheet is twelve geometry formulas and three notes, and Circles is not one of them. It is one of the 24 things the test expects you to know without being given them. See what the reference sheet does cover at /blog/sat-math-reference-sheet.