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MATH / Let ABCD be a rectangle. Let E and F be points on BC and CD, respectively, so that the areas of triangles ABE, ADF, and CEF are 8, 5, and 9, respectively. Find the area of rectangl…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let ABCDABCD be a rectangle. Let EE and FF be points on BCBC and CDCD, respectively, so that the areas of triangles ABEABE, ADFADF, and CEFCEF are 8, 5, and 9, respectively. Find the area of rectangle ABCDABCD. [asy] unitsize(1.5 cm); pair A, B, C, D, E, F; A = (0,2); B = (0,0); C = (3,0); D = (3,2); E = (3*B + 2*C)/5; F = (2*D + C)/3; draw(A--B--C--D--cycle); draw(A--E--F--cycle); label("AA", A, NW); label("BB", B, SW); label("CC", C, SE); label("DD", D, NE); label("EE", E, S); label("FF", F, dir(0)); label("88", (A + B + E)/3); label("55", (A + D + F)/3); label("99", (C + E + F)/3); [/asy]
Plain-text mathematical notation (without MathML)
Let ABCD be a rectangle.  Let E and F be points on BC and CD, respectively, so that the areas of triangles ABE, ADF, and CEF are 8, 5, and 9, respectively.  Find the area of rectangle ABCD.

[asy]
unitsize(1.5 cm);

pair A, B, C, D, E, F;

A = (0,2);
B = (0,0);
C = (3,0);
D = (3,2);
E = (3*B + 2*C)/5;
F = (2*D + C)/3;

draw(A--B--C--D--cycle);
draw(A--E--F--cycle);

label("A", A, NW);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("E", E, S);
label("F", F, dir(0));
label("8", (A + B + E)/3);
label("5", (A + D + F)/3);
label("9", (C + E + F)/3);
[/asy]
Original LaTeX notation
Let $ABCD$ be a rectangle.  Let $E$ and $F$ be points on $BC$ and $CD$, respectively, so that the areas of triangles $ABE$, $ADF$, and $CEF$ are 8, 5, and 9, respectively.  Find the area of rectangle $ABCD$.

[asy]
unitsize(1.5 cm);

pair A, B, C, D, E, F;

A = (0,2);
B = (0,0);
C = (3,0);
D = (3,2);
E = (3*B + 2*C)/5;
F = (2*D + C)/3;

draw(A--B--C--D--cycle);
draw(A--E--F--cycle);

label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, NE);
label("$E$", E, S);
label("$F$", F, dir(0));
label("$8$", (A + B + E)/3);
label("$5$", (A + D + F)/3);
label("$9$", (C + E + F)/3);
[/asy]

level

Level 4

type

Geometry

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Source and history

task_id
bb1a8267-9693-5a79-8fbd-2a414550cb69
task_key
geometry--test--bb1a8267-9693-5a79-8fbd-2a414550cb69
task_revision_id
1
config / split
geometry / test

Official source

initial import