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MATH / In the diagram, four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of △ABC, as shown. [asy] size(200); pair A, B, C, P, Q, R, S; R=(0,…

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problem

In the diagram, four circles of radius 1 with centres PP, QQ, RR, and SS are tangent to one another and to the sides of △ABC\triangle ABC, as shown. [asy] size(200); pair A, B, C, P, Q, R, S; R=(0,0); Q=(-2,0); S=(2,0); P=(1,1.732); B=(-5.73,-1); C=(3.732,-1); A=(1.366,3.098); draw(A--B--C--A); draw(circle(P, 1)); draw(circle(Q, 1)); draw(circle(R, 1)); draw(circle(S, 1)); label("A", A, N); label("B", B, SW); label("C", C, SE); dot(P); dot(Q); dot(R); dot(S); label("P", P, N); label("Q", Q, SW); label("R", R, SW); label("S", S, SE); [/asy] What is the degree measure of the smallest angle in triangle PQSPQS?
Plain-text mathematical notation (without MathML)
In the diagram, four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of △ABC, as shown. [asy]
size(200);
pair A, B, C, P, Q, R, S;
R=(0,0);
Q=(-2,0);
S=(2,0);
P=(1,1.732);
B=(-5.73,-1);
C=(3.732,-1);
A=(1.366,3.098);
draw(A--B--C--A);
draw(circle(P, 1));
draw(circle(Q, 1));
draw(circle(R, 1));
draw(circle(S, 1));
label("A", A, N);
label("B", B, SW);
label("C", C, SE);
dot(P);
dot(Q);
dot(R);
dot(S);
label("P", P, N);
label("Q", Q, SW);
label("R", R, SW);
label("S", S, SE);
[/asy]


What is the degree measure of the smallest angle in triangle PQS?
Original LaTeX notation
In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown. [asy]
size(200);
pair A, B, C, P, Q, R, S;
R=(0,0);
Q=(-2,0);
S=(2,0);
P=(1,1.732);
B=(-5.73,-1);
C=(3.732,-1);
A=(1.366,3.098);
draw(A--B--C--A);
draw(circle(P, 1));
draw(circle(Q, 1));
draw(circle(R, 1));
draw(circle(S, 1));
label("A", A, N);
label("B", B, SW);
label("C", C, SE);
dot(P);
dot(Q);
dot(R);
dot(S);
label("P", P, N);
label("Q", Q, SW);
label("R", R, SW);
label("S", S, SE);
[/asy]


What is the degree measure of the smallest angle in triangle $PQS$?

level

Level 3

type

Geometry

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Source and history

task_id
5457d47a-e2e5-5103-b27e-5589d6f5a9bd
task_key
geometry--test--5457d47a-e2e5-5103-b27e-5589d6f5a9bd
task_revision_id
1
config / split
geometry / test

Official source

initial import