text: # 2000 AMC 10 Problems/Problem 7
  
  (diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
  
  ## Problem
  
  In rectangle $ABCD$, $AD=1$, $P$ is on $\overline{AB}$, and $\overline{DB}$ and $\overline{DP}$ trisect $\angle ADC$. What is the perimeter of $\triangle BDP$?
  
  $[asy] draw((0,2)--(3.4,2)--(3.4,0)--(0,0)--cycle); draw((0,0)--(1.3,2)); draw((0,0)--(3.4,2)); dot((0,0)); dot((0,2)); dot((3.4,2)); dot((3.4,0)); dot((1.3,2)); label("A",(0,2),NW); label("B",(3.4,2),NE); label("C",(3.4,0),SE); label("D",(0,0),SW); label("P",(1.3,2),N); [/asy]$
  
  $\textbf{(A)}\ 3+\frac{\sqrt{3}}{3} \qquad\textbf{(B)}\ 2+\frac{4\sqrt{3}}{3} \qquad\textbf{(C)}\ 2+2\sqrt{2} \qquad\textbf{(D)}\ \frac{3+3\sqrt{5}}{2} \qquad\textbf{(E)}\ 2+\frac{5\sqrt{3}}{3}$
  
  ## Solution
  
  $[asy] draw((0,2)--(3.4,2)--(3.4,0)--(0,0)--cycle); draw((0,0)--(1.3,2)); draw((0,0)--(3.4,2)); dot((0,0)); dot((0,2)); dot((3.4,2)); dot((3.4,0)); dot((1.3,2)); label("A",(0,2),NW); label("B",(3.4,2),NE); label("C",(3.4,0),SE); label("D",(0,0),SW); label("P",(1.3,2),N); label("1",(0,1),W); label("2",(1.7,1),SE); label("\frac{\sqrt{3}}{3}",(0.65,2),N); label("\frac{2\sqrt{3}}{3}",(0.85,1),NW); label("\frac{2\sqrt{3}}{3}",(2.35,2),N); [/asy]$
  
  $AD=1$.
  
  Since $\angle ADC$ is trisected, $\angle ADP= \angle PDB= \angle BDC=30^\circ$.
  
  Thus, $PD=\frac{2\sqrt{3}}{3}$
  
  $DB=2$
  
  $BP=\sqrt{3}-\frac{\sqrt{3}}{3}=\frac{2\sqrt{3}}{3}$.
  
  Adding, we get $\boxed{\textbf{(B) } 2+\frac{4\sqrt{3}}{3}}$.
explanation: The document contains a math problem from the 2000 AMC 10 competition, specifically Problem 7. It includes a detailed problem statement with a diagram and a solution that calculates the perimeter of triangle BDP. The solution provides step-by-step reasoning and the final answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 2000 AMC 10 Problems/Problem 7
  
  (diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
  
  ## Problem
  
  In rectangle $ABCD$, $AD=1$, $P$ is on $\overline{AB}$, and $\overline{DB}$ and $\overline{DP}$ trisect $\angle ADC$. What is the perimeter of $\triangle BDP$?
  
  $[asy] draw((0,2)--(3.4,2)--(3.4,0)--(0,0)--cycle); draw((0,0)--(1.3,2)); draw((0,0)--(3.4,2)); dot((0,0)); dot((0,2)); dot((3.4,2)); dot((3.4,0)); dot((1.3,2)); label("A",(0,2),NW); label("B",(3.4,2),NE); label("C",(3.4,0),SE); label("D",(0,0),SW); label("P",(1.3,2),N); [/asy]$
  
  $\textbf{(A)}\ 3+\frac{\sqrt{3}}{3} \qquad\textbf{(B)}\ 2+\frac{4\sqrt{3}}{3} \qquad\textbf{(C)}\ 2+2\sqrt{2} \qquad\textbf{(D)}\ \frac{3+3\sqrt{5}}{2} \qquad\textbf{(E)}\ 2+\frac{5\sqrt{3}}{3}$
  
  ## Solution
  
  $[asy] draw((0,2)--(3.4,2)--(3.4,0)--(0,0)--cycle); draw((0,0)--(1.3,2)); draw((0,0)--(3.4,2)); dot((0,0)); dot((0,2)); dot((3.4,2)); dot((3.4,0)); dot((1.3,2)); label("A",(0,2),NW); label("B",(3.4,2),NE); label("C",(3.4,0),SE); label("D",(0,0),SW); label("P",(1.3,2),N); label("1",(0,1),W); label("2",(1.7,1),SE); label("\frac{\sqrt{3}}{3}",(0.65,2),N); label("\frac{2\sqrt{3}}{3}",(0.85,1),NW); label("\frac{2\sqrt{3}}{3}",(2.35,2),N); [/asy]$
  
  $AD=1$.
  
  Since $\angle ADC$ is trisected, $\angle ADP= \angle PDB= \angle BDC=30^\circ$.
  
  Thus, $PD=\frac{2\sqrt{3}}{3}$
  
  $DB=2$
  
  $BP=\sqrt{3}-\frac{\sqrt{3}}{3}=\frac{2\sqrt{3}}{3}$.
  
  Adding, we get $\boxed{\textbf{(B) } 2+\frac{4\sqrt{3}}{3}}$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem from the 2000 AMC 10 competition, specifically Problem 7. It includes a detailed problem statement with a diagram and a solution that calculates the perimeter of triangle BDP. The solution provides step-by-step reasoning and the final answer.
  
  Label: question-and-answer