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# Halp

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An equilateral triangle is constructed on each side of a square with side length 2 as shown below. The four outer vertices are then joined to form a large square. Find the side length of the large square.

[asy]
unitsize(1 cm);

draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
draw((1,1)--((1,1) + 2*dir(120))--(-1,1));
draw(rotate(90)*((1,1)--((1,1) + 2*dir(120))--(-1,1)));
draw(rotate(180)*((1,1)--((1,1) + 2*dir(120))--(-1,1)));
draw(rotate(270)*((1,1)--((1,1) + 2*dir(120))--(-1,1)));
draw((1 + sqrt(3))*dir(0)--(1 + sqrt(3))*dir(90)--(1 + sqrt(3))*dir(180)--(1 + sqrt(3))*dir(270)--cycle);

label("\$2\$", (0,1), S);
[/asy]

Apr 8, 2023

#1
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The equilateral triangle has side length sqrt(3), so the four outer vertices of the large square are sqrt(3)​ units away from each other. The large square has an inscribed circle, which is tangent to all four sides of the square. The radius of the inscribed circle is 2/sqrt(3)​, so the diameter of the inscribed circle is 2*sqrt(3)​. The side length of the large square is equal to the diameter of the inscribed circle plus the distance between the outer vertices, which is sqrt(3) + 2.

Apr 8, 2023
#2
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I'm sorry but the side lenghts of the equalitwral triangle are 2 and the answer is supposst to be somt like this 1sqrt2, In that format,

Guest Apr 9, 2023