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MATH / In the diagram, K, O and M are the centers of the three semi-circles. Also, OC=32 and CB=36. [asy] pair A, K, O, C, M, B, X, Y, Z, J, T; O=(0,0); C=(32,0); M=(50,0); B=(68,0); A=(-…

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problem

In the diagram, KK, OO and MM are the centers of the three semi-circles. Also, OC=32OC = 32 and CB=36CB = 36. [asy] pair A, K, O, C, M, B, X, Y, Z, J, T; O=(0,0); C=(32,0); M=(50,0); B=(68,0); A=(-68,0); K=(A+C)/2; X=(0,68); Y=(-18,50); Z=(50,18); J=(7,43.3); T=(59,15.6); path nom, bigc, middlec, smallc; nom=A--B--(100,100)--(-100,100)--cycle; bigc=A..X..B--cycle; middlec=A..Y..C--cycle; smallc=C..Z..B--cycle; fill(bigc, gray(.5)); fill(middlec, white); fill(smallc, white); draw(smallc); draw(middlec); draw(bigc); draw(A--B); label("A", A, S); label("K", K, S); label("O", O, S); label("M", M, S); label("C", C, S); label("B", B, S); label("S", J, SW); label("E", T, SW); label("ll", (.9(J-T)+J), NW); draw((.9(J-T)+J)--(.5(T-J)+T)); dot(K); dot(O); dot(M); dot(J); dot(T); [/asy] Line ll is drawn to touch the smaller semi-circles at points SS and EE so that KSKS and MEME are both perpendicular to ll. Determine the area of quadrilateral KSEMKSEM.
Plain-text mathematical notation (without MathML)
In the diagram, K, O and M are the centers of the three semi-circles. Also, OC=32 and CB=36.

[asy]
pair A, K, O, C, M, B, X, Y, Z, J, T;
O=(0,0);
C=(32,0);
M=(50,0);
B=(68,0);
A=(-68,0);
K=(A+C)/2;
X=(0,68);
Y=(-18,50);
Z=(50,18);
J=(7,43.3);
T=(59,15.6);
path nom, bigc, middlec, smallc;
nom=A--B--(100,100)--(-100,100)--cycle;
bigc=A..X..B--cycle;
middlec=A..Y..C--cycle;
smallc=C..Z..B--cycle;
fill(bigc, gray(.5));
fill(middlec, white);
fill(smallc, white);
draw(smallc);
draw(middlec);
draw(bigc);
draw(A--B);
label("A", A, S);
label("K", K, S);
label("O", O, S);
label("M", M, S);
label("C", C, S);
label("B", B, S);
label("S", J, SW);
label("E", T, SW);
label("l", (.9(J-T)+J), NW);
draw((.9(J-T)+J)--(.5(T-J)+T));
dot(K);
dot(O);
dot(M);
dot(J);
dot(T);
[/asy] Line l is drawn to touch the smaller semi-circles at points S and E so that KS and ME are both perpendicular to l.  Determine the area of quadrilateral KSEM.
Original LaTeX notation
In the diagram, $K$, $O$ and $M$ are the centers of the three semi-circles. Also, $OC = 32$ and $CB = 36$.

[asy]
pair A, K, O, C, M, B, X, Y, Z, J, T;
O=(0,0);
C=(32,0);
M=(50,0);
B=(68,0);
A=(-68,0);
K=(A+C)/2;
X=(0,68);
Y=(-18,50);
Z=(50,18);
J=(7,43.3);
T=(59,15.6);
path nom, bigc, middlec, smallc;
nom=A--B--(100,100)--(-100,100)--cycle;
bigc=A..X..B--cycle;
middlec=A..Y..C--cycle;
smallc=C..Z..B--cycle;
fill(bigc, gray(.5));
fill(middlec, white);
fill(smallc, white);
draw(smallc);
draw(middlec);
draw(bigc);
draw(A--B);
label("A", A, S);
label("K", K, S);
label("O", O, S);
label("M", M, S);
label("C", C, S);
label("B", B, S);
label("S", J, SW);
label("E", T, SW);
label("$l$", (.9(J-T)+J), NW);
draw((.9(J-T)+J)--(.5(T-J)+T));
dot(K);
dot(O);
dot(M);
dot(J);
dot(T);
[/asy] Line $l$ is drawn to touch the smaller semi-circles at points $S$ and $E$ so that $KS$ and $ME$ are both perpendicular to $l$.  Determine the area of quadrilateral $KSEM$.

level

Level 5

type

Geometry

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Source and history

task_id
3074641b-6599-5d52-9698-025e18c7d4ef
task_key
geometry--test--3074641b-6599-5d52-9698-025e18c7d4ef
task_revision_id
1
config / split
geometry / test

Official source

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