SAT Math formula sheet · all 24 · Geometry and Trigonometry · #20 of 24

Similar figures on the SAT: How It Is Tested

Lines, angles, and triangles

Similar Figures diagram for the SAT: Similar figures are very common on the test, and it is super important that you always remember the first two properties
What it is really testingSimilar figures are very common on the test, and it is super important that you always remember the first two properties. The third property, the one about trig, is not always used, but it is nice to know.

The question from the sheet

Hard
SAT Geometry and Trigonometryanswer C
LMRPQNote: Figure not drawn to scale.In the figure, LQ‾\overline{LQ} intersects MP‾\overline{MP} at point RR, and LM‾\overline{LM} is parallel to PQ‾\overline{PQ}. The lengths of MR‾\overline{MR}, LR‾\overline{LR}, and RP‾\overline{RP} are 77, 99, and 1212, respectively. What is the length of LQ‾\overline{LQ}?
A2121
B1087\frac{108}{7}
✓1717\frac{171}{7}
D553\frac{55}{3}

How the diagram applies

  1. LM∥PQLM \parallel PQ, so the alternate interior angles are equal (∠L=∠Q\angle L = \angle Q, ∠M=∠P\angle M = \angle P) and the vertical angles at RR are equal. Triangles LMRLMR and QPRQPR are similar.
  2. Match sides by the angles they sit across from: MRMR corresponds to PRPR, so the scale factor is 127\tfrac{12}{7}. Then LRLR corresponds to QRQR: QR=9⋅127=1087QR = 9 \cdot \tfrac{12}{7} = \tfrac{108}{7}.
  3. LQ=LR+RQ=9+1087=1717LQ = LR + RQ = 9 + \tfrac{108}{7} = \tfrac{171}{7}.

Answer: C

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Common Questions

It comes up as an SAT lines, angles, and triangles question. Similar figures are very common on the test, and it is super important that you always remember the first two properties. The third property, the one about trig, is not always used, but it is nice to know.

LM PQ, so the alternate interior angles are equal (∠ L = ∠ Q, ∠ M = ∠ P) and the vertical angles at R are equal. Triangles LMR and QPR are similar. Match sides by the angles they sit across from: MR corresponds to PR, so the scale factor is 127. Then LR corresponds to QR: QR = 9 × 127 = 1087. LQ = LR + RQ = 9 + 1087 = 1717.

No. The SAT reference sheet is twelve geometry formulas and three notes, and Similar Figures 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.