An architect planned to construct two similar stone pyramid structures in a park. The figure below shows the front view of the pyramids in her plan, but there is an error in the dimensions:Two similar scalene triangles PQR and ABC with angle P congruent to angle A, angle Q congruent to angle B, and QR and BC as the bases of the triangles. The length of PQ is 6 feet, the length of QR is 9.5 feet, and the length of RP is 7.5 feet. The length of AB is 4 feet, the length of BC is 7 feet, and the length of CA is 5 feet.Which of the following changes should she make to the length of side RQ to correct her error? Change the length of side RQ to 9 feet Change the length of side RQ to 10.5 feet Change the length of side RQ to 8 feet Change the length of side RQ to 11.5 feet
Accepted Solution
A:
Answer:The correct option is 2. She can change the length of side RQ to 10.5 feet to correct her error.Step-by-step explanation:In triangle PQR and ABC,[tex]\angle P=\angle A[/tex] (Given)[tex]\angle Q=\angle B[/tex] (Given)By AA rule of similarity,[tex]\triangle PQR=\triangle ABC[/tex]The corresponding sides of similar triangles are proportional.[tex]\frac{PQ}{AB}=\frac{RQ}{CB}=\frac{PR}{AC}[/tex][tex]\frac{PQ}{AB}=\frac{6}{4}=1.5[/tex][tex]\frac{RQ}{CB}=\frac{9.5}{7}=1.35714285714[/tex][tex]\frac{PQ}{AB}neq \frac{RQ}{CB}[/tex]It means there is an error in the dimensions. Let the new length of RQ be x.[tex]\frac{PQ}{AB}=\frac{RQ}{CB}[/tex][tex]\frac{6}{4}=\frac{x}{7}[/tex]Multiply both sides by 7.[tex]\frac{6\times 7}{4}=x[/tex][tex]\frac{42}{4}=x[/tex][tex]10.50=x[/tex]Therefore the length of RQ must be 10.50. The correct option is 2.