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problem
In the diagram, four squares of side length 2 are placed in the corners of a square of side length 6. Each of the points , , , and is a vertex of one of the small squares. Square can be constructed with sides passing through , , , and . What is the maximum possible distance from to ? [asy]
path square = scale(2) * unitsquare;
draw(square); draw(shift(4) * square); draw(shift(4, 4) * square); draw(shift(0, 4) * square);
draw((2, 0)--(4, 0)); draw((0, 2)--(0, 4)); draw((6, 2)--(6, 4)); draw((2, 6)--(4, 6));
pair a = shift(3, 4) * dir(135);
pair b = shift(4, 3) * dir(45);
pair c = shift(3, 2) * dir(-45);
pair d = shift(2, 3) * dir(-135);
draw(a--b--c--d--cycle);
label("", (1, 6), N); label("", (3, 6), N); label("", (5, 6), N);
label("", (6, 5), E); label("", (6, 3), E); label("", (6, 1), E);
label("", (2, 4), NW); label("", (4, 4), NE); label("", (4, 2), SE); label("", (2, 2), SW);
label("", a, N); label("", b, E); label("", c, S); label("", d, W); label("", (6, 0), SE);
[/asy]
Plain-text mathematical notation (without MathML)
In the diagram, four squares of side length 2 are placed in the corners of a square of side length 6. Each of the points W, X, Y, and Z is a vertex of one of the small squares. Square ABCD can be constructed with sides passing through W, X, Y, and Z. What is the maximum possible distance from A to P? [asy]
path square = scale(2) * unitsquare;
draw(square); draw(shift(4) * square); draw(shift(4, 4) * square); draw(shift(0, 4) * square);
draw((2, 0)--(4, 0)); draw((0, 2)--(0, 4)); draw((6, 2)--(6, 4)); draw((2, 6)--(4, 6));
pair a = shift(3, 4) * dir(135);
pair b = shift(4, 3) * dir(45);
pair c = shift(3, 2) * dir(-45);
pair d = shift(2, 3) * dir(-135);
draw(a--b--c--d--cycle);
label("2", (1, 6), N); label("2", (3, 6), N); label("2", (5, 6), N);
label("2", (6, 5), E); label("2", (6, 3), E); label("2", (6, 1), E);
label("W", (2, 4), NW); label("X", (4, 4), NE); label("Y", (4, 2), SE); label("Z", (2, 2), SW);
label("A", a, N); label("B", b, E); label("C", c, S); label("D", d, W); label("P", (6, 0), SE);
[/asy]Original LaTeX notation
In the diagram, four squares of side length 2 are placed in the corners of a square of side length 6. Each of the points $W$, $X$, $Y$, and $Z$ is a vertex of one of the small squares. Square $ABCD$ can be constructed with sides passing through $W$, $X$, $Y$, and $Z$. What is the maximum possible distance from $A$ to $P$? [asy]
path square = scale(2) * unitsquare;
draw(square); draw(shift(4) * square); draw(shift(4, 4) * square); draw(shift(0, 4) * square);
draw((2, 0)--(4, 0)); draw((0, 2)--(0, 4)); draw((6, 2)--(6, 4)); draw((2, 6)--(4, 6));
pair a = shift(3, 4) * dir(135);
pair b = shift(4, 3) * dir(45);
pair c = shift(3, 2) * dir(-45);
pair d = shift(2, 3) * dir(-135);
draw(a--b--c--d--cycle);
label("$2$", (1, 6), N); label("$2$", (3, 6), N); label("$2$", (5, 6), N);
label("$2$", (6, 5), E); label("$2$", (6, 3), E); label("$2$", (6, 1), E);
label("$W$", (2, 4), NW); label("$X$", (4, 4), NE); label("$Y$", (4, 2), SE); label("$Z$", (2, 2), SW);
label("$A$", a, N); label("$B$", b, E); label("$C$", c, S); label("$D$", d, W); label("$P$", (6, 0), SE);
[/asy]level
Level 5
type
Geometry
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