Difference between revisions of "2000 AMC 10 Problems/Problem 7"

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Revision as of 20:39, 18 January 2021

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]

$\mathrm{(A)}\ 3+\frac{\sqrt{3}}{3} \qquad\mathrm{(B)}\ 2+\frac{4\sqrt{3}}{3} \qquad\mathrm{(C)}\ 2+2\sqrt{2} \qquad\mathrm{(D)}\ \frac{3+3\sqrt{5}}{2} \qquad\mathrm{(E)}\ 2+\frac{5\sqrt{3}}{3}$

Solution

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);
 (Error compiling LaTeX. 5559f77f01b77e58da076397f8db0038473786f3.asy: 9.10: syntax error
error: could not load module '5559f77f01b77e58da076397f8db0038473786f3.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 $\boxed{\textbf{(B) }  2+\frac{4\sqrt{3}}{3}}$.

See Also

2000 AMC 10 (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 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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