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Difficulty: EasyTriangle Congruence, Similarity, and Theorems

In the triangles PQRPQR and STUSTU shown, ∠P≅∠S\angle P \cong \angle S and ∠Q≅∠T\angle Q \cong \angle T. If the length of PQPQ is 1010, the length of QRQR is 1515, and the length of STST is 66, what is the length of TUTU?

  1. A
    4
  2. 9Answer
  3. C
    11
  4. D
    25

Answer

9
Since two angles of triangle PQRPQR are congruent to two angles of triangle STUSTU, the triangles are similar by the Angle-Angle similarity theorem. Therefore, the ratio of corresponding side lengths is constant, which gives the equation PQST=QRTU\frac{PQ}{ST} = \frac{QR}{TU}. Substituting the given values yields 106=15TU\frac{10}{6} = \frac{15}{TU}. Solving for TUTU gives 10×TU=9010 \times TU = 90, which simplifies to TU=9TU = 9.

Step-by-Step Solution

1
Identify the relationship between the two triangles.
The triangles are similar (△PQR∼△STU\triangle PQR \sim \triangle STU).
Since two angles of triangle PQRPQR are congruent to two angles of triangle STUSTU, the triangles are similar by the Angle-Angle (AA) similarity theorem.
2
Set up a proportion using corresponding side lengths.
PQST=QRTU\frac{PQ}{ST} = \frac{QR}{TU}
Corresponding sides of similar triangles are proportional.
3
Substitute the known values and solve for the unknown side length.
TU=9TU = 9
Substituting the values gives 106=15TU\frac{10}{6} = \frac{15}{TU}. Cross-multiplying yields 10×TU=9010 \times TU = 90, which simplifies to TU=9TU = 9.

Key Concept

Triangle Similarity (AA Postulate)

Alternative Method

Find the scale factor from triangle PQRPQR to triangle STUSTU, which is STPQ=610=0.6\frac{ST}{PQ} = \frac{6}{10} = 0.6. Then, multiply the corresponding side QRQR by this scale factor: 15×0.6=915 \times 0.6 = 9.
Estimated Time:45s
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Topics

Triangle Congruence, Similarity, and Theorems

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