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MATH / In the diagram, square ABCD has sides of length 4, and △ABE is equilateral. Line segments BE and AC intersect at P. Point Q is on BC so that PQ is perpendicular to BC and PQ=x. [as…

Problem

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problem

In the diagram, square ABCDABCD has sides of length 4, and △ABE\triangle ABE is equilateral. Line segments BEBE and ACAC intersect at PP. Point QQ is on BCBC so that PQPQ is perpendicular to BCBC and PQ=xPQ=x. [asy] pair A, B, C, D, E, P, Q; A=(0,0); B=(4,0); C=(4,-4); D=(0,-4); E=(2,-3.464); P=(2.535,-2.535); Q=(4,-2.535); draw(A--B--C--D--A--E--B); draw(A--C); draw(P--Q, dashed); label("AA", A, NW); label("BB", B, NE); label("CC", C, SE); label("DD", D, SW); label("EE", E, S); label("PP", P, W); label("QQ", Q, dir(0)); label("xx", (P+Q)/2, N); label("4", (A+B)/2, N); [/asy] Find the value of xx in simplest radical form.
Plain-text mathematical notation (without MathML)
In the diagram, square ABCD has sides of length 4, and △ABE is equilateral. Line segments BE and AC intersect at P. Point Q is on BC so that PQ is perpendicular  to BC and PQ=x. [asy]
pair A, B, C, D, E, P, Q;
A=(0,0);
B=(4,0);
C=(4,-4);
D=(0,-4);
E=(2,-3.464);
P=(2.535,-2.535);
Q=(4,-2.535);
draw(A--B--C--D--A--E--B);
draw(A--C);
draw(P--Q, dashed);
label("A", A, NW);
label("B", B, NE);
label("C", C, SE);
label("D", D, SW);
label("E", E, S);
label("P", P, W);
label("Q", Q, dir(0));
label("x", (P+Q)/2, N);
label("4", (A+B)/2, N);
[/asy]

Find the value of x in simplest radical form.
Original LaTeX notation
In the diagram, square $ABCD$ has sides of length 4, and $\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P$. Point $Q$ is on $BC$ so that $PQ$ is perpendicular  to $BC$ and $PQ=x$. [asy]
pair A, B, C, D, E, P, Q;
A=(0,0);
B=(4,0);
C=(4,-4);
D=(0,-4);
E=(2,-3.464);
P=(2.535,-2.535);
Q=(4,-2.535);
draw(A--B--C--D--A--E--B);
draw(A--C);
draw(P--Q, dashed);
label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, SE);
label("$D$", D, SW);
label("$E$", E, S);
label("$P$", P, W);
label("$Q$", Q, dir(0));
label("$x$", (P+Q)/2, N);
label("4", (A+B)/2, N);
[/asy]

Find the value of $x$ in simplest radical form.

level

Level 5

type

Geometry

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Source and history

task_id
0f13c388-8f98-54be-b2d0-4c38d0ea01f5
task_key
geometry--test--0f13c388-8f98-54be-b2d0-4c38d0ea01f5
task_revision_id
1
config / split
geometry / test

Official source

initial import