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MATH / In the diagram, R is on QS and QR=8. Also, PR=12, ∠PRQ=120^(∘), and ∠RPS=90^(∘). What is the area of △QPS? [asy] //C19 size(10cm); // ADJUST import olympiad; // Point coordinates p…

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problem

In the diagram, RR is on QSQS and QR=8QR=8. Also, PR=12PR=12, ∠PRQ=120∘\angle PRQ=120^\circ, and ∠RPS=90∘\angle RPS = 90^\circ. What is the area of △QPS\triangle QPS? [asy] //C19 size(10cm); // ADJUST import olympiad; // Point coordinates pair r = (0, 0); pair q = (-8, 0); pair p = 12 * dir(60); pair s = (24, 0); // draw lines draw(p--s--r--p--q--r); // labels label("RR", r, S); label("QQ", q, SW); label("SS", s, SE); label("PP", p, N); label("1212", r + (p - r) / 2, SE); label("88", r + (q - r) / 2, S); label("120∘120^\circ", r, NW + 0.2 * E); markscalefactor = 0.08; draw(rightanglemark(r, p, s)); [/asy]
Plain-text mathematical notation (without MathML)
In the diagram, R is on QS and QR=8. Also, PR=12, ∠PRQ=120^(∘), and ∠RPS=90^(∘). What is the area of △QPS? [asy]
//C19
size(10cm); // ADJUST
import olympiad;

// Point coordinates
pair r = (0, 0);
pair q = (-8, 0);
pair p = 12 * dir(60);
pair s = (24, 0);

// draw lines
draw(p--s--r--p--q--r);

// labels
label("R", r, S);
label("Q", q, SW);
label("S", s, SE);
label("P", p, N);
label("12", r + (p - r) / 2, SE);
label("8", r + (q - r) / 2, S);
label("120^(∘)", r, NW + 0.2 * E);

markscalefactor = 0.08;
draw(rightanglemark(r, p, s));
[/asy]
Original LaTeX notation
In the diagram, $R$ is on $QS$ and $QR=8$. Also, $PR=12$, $\angle PRQ=120^\circ$, and $\angle RPS = 90^\circ$. What is the area of $\triangle QPS$? [asy]
//C19
size(10cm); // ADJUST
import olympiad;

// Point coordinates
pair r = (0, 0);
pair q = (-8, 0);
pair p = 12 * dir(60);
pair s = (24, 0);

// draw lines
draw(p--s--r--p--q--r);

// labels
label("$R$", r, S);
label("$Q$", q, SW);
label("$S$", s, SE);
label("$P$", p, N);
label("$12$", r + (p - r) / 2, SE);
label("$8$", r + (q - r) / 2, S);
label("$120^\circ$", r, NW + 0.2 * E);

markscalefactor = 0.08;
draw(rightanglemark(r, p, s));
[/asy]

level

Level 4

type

Geometry

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Source and history

task_id
18e68315-0399-5873-831b-1f8de4ef3621
task_key
geometry--test--18e68315-0399-5873-831b-1f8de4ef3621
task_revision_id
1
config / split
geometry / test

Official source

initial import