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MATH / In regular pentagon PQRST, X is the midpoint of segment ST. What is the measure of angle XQS, in degrees? [asy] size(101); picture p; pair P = (0,0); pair Q = (-cos(108),sin(108));…

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problem

In regular pentagon PQRSTPQRST, XX is the midpoint of segment STST. What is the measure of angle XQS,XQS, in degrees? [asy] size(101); picture p; pair P = (0,0); pair Q = (-cos(108),sin(108)); pair R = Q + expi(36*pi/180); pair T = (1,0); pair S1 = T + (cos(108),sin(108)); draw(p,(P--Q--R--S1--T--cycle),linewidth(1)); pair X = (S1+T)/2; label(p,rotate(20)*"PP",P,rotate(20)*W,fontsize(10pt)); label(p,rotate(20)*"QQ",Q,rotate(20)*NW,fontsize(10pt)); label(p,rotate(20)*"RR",R,S,fontsize(10pt)); label(p,rotate(20)*"SS",S1,rotate(20)*E,fontsize(10pt)); label(p,rotate(20)*"TT",T,rotate(20)*N,fontsize(10pt));label(p,rotate(20)*"XX",X,rotate(-40)*NE,fontsize(10pt)); dot(p,X); add(rotate(-20)*p); [/asy]
Plain-text mathematical notation (without MathML)
In regular pentagon PQRST, X is the midpoint of segment ST. What is the measure of angle XQS, in degrees?

[asy]
size(101);
picture p;
pair P = (0,0); pair Q = (-cos(108),sin(108)); pair R = Q + expi(36*pi/180); pair T = (1,0); pair S1 = T + (cos(108),sin(108));
draw(p,(P--Q--R--S1--T--cycle),linewidth(1)); pair X = (S1+T)/2;
label(p,rotate(20)*"P",P,rotate(20)*W,fontsize(10pt)); label(p,rotate(20)*"Q",Q,rotate(20)*NW,fontsize(10pt)); label(p,rotate(20)*"R",R,S,fontsize(10pt)); label(p,rotate(20)*"S",S1,rotate(20)*E,fontsize(10pt)); label(p,rotate(20)*"T",T,rotate(20)*N,fontsize(10pt));label(p,rotate(20)*"X",X,rotate(-40)*NE,fontsize(10pt));
dot(p,X);
add(rotate(-20)*p);
[/asy]
Original LaTeX notation
In regular pentagon $PQRST$, $X$ is the midpoint of segment $ST$. What is the measure of angle $XQS,$ in degrees?

[asy]
size(101);
picture p;
pair P = (0,0); pair Q = (-cos(108),sin(108)); pair R = Q + expi(36*pi/180); pair T = (1,0); pair S1 = T + (cos(108),sin(108));
draw(p,(P--Q--R--S1--T--cycle),linewidth(1)); pair X = (S1+T)/2;
label(p,rotate(20)*"$P$",P,rotate(20)*W,fontsize(10pt)); label(p,rotate(20)*"$Q$",Q,rotate(20)*NW,fontsize(10pt)); label(p,rotate(20)*"$R$",R,S,fontsize(10pt)); label(p,rotate(20)*"$S$",S1,rotate(20)*E,fontsize(10pt)); label(p,rotate(20)*"$T$",T,rotate(20)*N,fontsize(10pt));label(p,rotate(20)*"$X$",X,rotate(-40)*NE,fontsize(10pt));
dot(p,X);
add(rotate(-20)*p);
[/asy]

level

Level 4

type

Geometry

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Source and history

task_id
cadd3b1b-961a-5b5d-87c4-ea2c99057c2b
task_key
geometry--test--cadd3b1b-961a-5b5d-87c4-ea2c99057c2b
task_revision_id
1
config / split
geometry / test

Official source

initial import