2000 AMC 10 Problems/Problem 7

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

$[asy] draw((0,2)--(3.4,2)--(3.4,0)--(0,0)--cycle); draw((0,0)--(1.3,2)); draw((0,0)--(3.4,2)); dot((0,0)); dot((0,2)); dot((3.4,2)); dot((3.4,0)); dot((1.3,2)); label("A",(0,2),NW); label("B",(3.4,2),NE); label("C",(3.4,0),SE); label("D",(0,0),SW); label("P",(1.3,2),N); label("1",(0,1),W); label("2",(1.7,1),SE); label("\frac{\sqrt{3}}{3}",(0.65,2),N); label("\frac{2\sqrt{3}}{3}",(0.85,1),NW); label("\frac{2\sqrt{3}}{3}",(2.35,2),N); label("\sqrt{3}",(1.7,0),S); label("2",(3,1),W); [/asy]$

$AD=1$.

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 $\boxed{\textbf{(B) } 2+\frac{4\sqrt{3}}{3}}$.

Solution 2

After computing $\overline{BP} = \frac{2\sqrt{3}}{3},$ observe that triangle $\triangle BPD$ is isosceles with $\angle DPB = \angle BPD.$ Therefore, using $120 - 30 - 30$ triangle properties, we see that the perimeter is just $(2+ \sqrt{3}) \cdot \frac{2\sqrt{3}}{3} = \boxed{\textbf{(B) } 2+\frac{4\sqrt{3}}{3}}.$

~Sliced_Bread

Video Solution by Daily Dose of Math

~Thesmartgreekmathdude

See Also

 2000 AMC 10 (Problems • Answer Key • Resources) Preceded byProblem 6 Followed byProblem 8 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions

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