Math: Geometry and Trigonometry

SAT Question of the Day for October 6, 2026

In triangle

PQR

, point

S

lies on

\overline{PQ}

and point

T

lies on

\overline{QR}

such that

\overline{ST}

is parallel to

\overline{PR}

. If

QS = 14

SP = 21

, and

ST = 18

, what is the length of

\overline{PR}

Correct answer: 45

Explanation

Since

\overline{ST}

is parallel to

\overline{PR}

, triangles

QST

and

QPR

are similar by the AA similarity criterion. The ratio of corresponding sides equals

\frac{QS}{QP} = \frac{14}{14 + 21} = \frac{14}{35} = \frac{2}{5}

. Therefore

\frac{ST}{PR} = \frac{2}{5}

, which gives

PR = 18 \times \frac{5}{2} = 45

. A common error is using the ratio

\frac{QS}{SP} = \frac{14}{21} = \frac{2}{3}

instead of

\frac{QS}{QP} = \frac{2}{5}

, which would incorrectly yield

PR = 18 \times \frac{3}{2} = 27

. Another error is applying the ratio in the wrong direction, computing

18 \times \frac{2}{5} = 7.2

instead of dividing.

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