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Question Watch Video Show Examples In the diagram below, bar(ST) is parallel to bar(PQ).RS=13.3,RT=6.7, and SP=10.7. Find the length of bar(TQ). Round your answer to the nearest tenth if necessary.

Question\newlineWatch Video\newlineShow Examples\newlineIn the diagram below, ST‾ \overline{S T} is parallel to PQ‾.RS=13.3,RT=6.7 \overline{P Q} . R S=13.3, R T=6.7 , and SP=10.7 S P=10.7 . Find the length of TQ‾ \overline{T Q} . Round your answer to the nearest tenth if necessary.

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Full solution

Q. Question\newlineWatch Video\newlineShow Examples\newlineIn the diagram below, ST‾ \overline{S T} is parallel to PQ‾.RS=13.3,RT=6.7 \overline{P Q} . R S=13.3, R T=6.7 , and SP=10.7 S P=10.7 . Find the length of TQ‾ \overline{T Q} . Round your answer to the nearest tenth if necessary.
  1. Identify Similar Triangles: Since ST‾\overline{ST} is parallel to PQ‾\overline{PQ}, and we have a transversal line that intersects them, we can use the properties of similar triangles to find the length of TQ‾\overline{TQ}. The triangles RSTRST and RPQRPQ are similar by the AA (Angle-Angle) similarity postulate because they have two pairs of corresponding angles that are congruent.
  2. Set Up Proportion: To find the length of bar(TQ), we will set up a proportion using the corresponding sides of the similar triangles RST and RPQ. The proportion is RSSP=RTTQ.\frac{RS}{SP} = \frac{RT}{TQ}.
  3. Substitute Known Lengths: Substitute the known lengths into the proportion: 13.310.7=6.7TQ\frac{13.3}{10.7} = \frac{6.7}{TQ}.
  4. Cross-Multiply to Solve: Solve for TQTQ by cross-multiplying: 13.3×TQ=6.7×10.713.3 \times TQ = 6.7 \times 10.7.
  5. Perform Multiplication: Perform the multiplication: 13.3×TQ=71.6913.3 \times TQ = 71.69.
  6. Isolate TQ: Divide both sides by 13.313.3 to isolate TQ: TQ=71.6913.3TQ = \frac{71.69}{13.3}.
  7. Calculate TQ: Calculate the value of TQ: TQ≈5.4TQ \approx 5.4 (rounded to the nearest tenth).

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