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MATH / Given regular pentagon ABCDE, a circle can be drawn that is tangent to (DC)¯ at D and to (AB)¯ at A. In degrees, what is the measure of minor arc AD? [asy]import olympiad; import g…

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problem

Given regular pentagon ABCDE,ABCDE, a circle can be drawn that is tangent to DC¯\overline{DC} at DD and to AB¯\overline{AB} at A.A. In degrees, what is the measure of minor arc ADAD? [asy]import olympiad; import geometry; size(100); defaultpen(linewidth(0.8)); pair[] pentagon = new pair[5]; pentagon[0] = dir(36); pentagon.cyclic=true; for(int i = 1; i < 6; ++i){ pentagon[i] = dir(72*i + 36); draw(pentagon[i - 1]--pentagon[i]); } dot("CC",pentagon[0],NE); dot("DD",pentagon[1],N); dot("EE",pentagon[2],W); dot("AA",pentagon[3],S); dot("BB",pentagon[4],SE); line x = line(pentagon[1],pentagon[1] + dir(-108)); line x2 = line(origin,pentagon[2]); pair ccenter = coordinates(intersectionpoint(x,x2)); draw(Circle(ccenter,arclength(ccenter--pentagon[1]))); [/asy]
Plain-text mathematical notation (without MathML)
Given regular pentagon ABCDE, a circle can be drawn that is tangent to (DC)¯ at D and to (AB)¯ at A. In degrees, what is the measure of minor arc AD? [asy]import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
pair[] pentagon = new pair[5];
pentagon[0] = dir(36);
pentagon.cyclic=true;
for(int i = 1; i < 6; ++i){
 
pentagon[i] = dir(72*i + 36);

draw(pentagon[i - 1]--pentagon[i]);
}
dot("C",pentagon[0],NE);
dot("D",pentagon[1],N);
dot("E",pentagon[2],W);
dot("A",pentagon[3],S);
dot("B",pentagon[4],SE);
line x = line(pentagon[1],pentagon[1] + dir(-108));
line x2 = line(origin,pentagon[2]);
pair ccenter = coordinates(intersectionpoint(x,x2));
draw(Circle(ccenter,arclength(ccenter--pentagon[1])));
[/asy]
Original LaTeX notation
Given regular pentagon $ABCDE,$ a circle can be drawn that is tangent to $\overline{DC}$ at $D$ and to $\overline{AB}$ at $A.$ In degrees, what is the measure of minor arc $AD$? [asy]import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
pair[] pentagon = new pair[5];
pentagon[0] = dir(36);
pentagon.cyclic=true;
for(int i = 1; i < 6; ++i){
 
pentagon[i] = dir(72*i + 36);

draw(pentagon[i - 1]--pentagon[i]);
}
dot("$C$",pentagon[0],NE);
dot("$D$",pentagon[1],N);
dot("$E$",pentagon[2],W);
dot("$A$",pentagon[3],S);
dot("$B$",pentagon[4],SE);
line x = line(pentagon[1],pentagon[1] + dir(-108));
line x2 = line(origin,pentagon[2]);
pair ccenter = coordinates(intersectionpoint(x,x2));
draw(Circle(ccenter,arclength(ccenter--pentagon[1])));
[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
92b9e004-1256-51e3-b2b1-bf23cfc7c602
task_key
geometry--test--92b9e004-1256-51e3-b2b1-bf23cfc7c602
task_revision_id
1
config / split
geometry / test

Official source

initial import