document.write( "Question 1057878: A side of an equilateral triangle is the diameter of a semi-circle. If the radius of the semi-circle is 1, find the area that is inside the triangle but outside the semi-circle. \n" ); document.write( "

Algebra.Com's Answer #673015 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A side of an equilateral triangle is the diameter of a semi-circle. If the radius of the semi-circle is 1,
\n" ); document.write( "find the area that is inside the triangle but outside the semi-circle.
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document.write( "The plot is shown in Figure 1 on the right.\r\n" );
document.write( "The point O is the center of the (semi) circle.\r\n" );
document.write( "The diameter is AB and the equilateral triangle is ABC.   \r\n" );
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document.write( "The points D and E are the intersection points of the   \r\n" );
document.write( "semicircle with the lateral sides of the triangle ABC.\r\n" );
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document.write( "The problem asks to find the area of a curved shape CDE.\r\n" );
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document.write( "Let us draw the radii OD and OE (Figure 2).\r\n" );
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document.write( "The triangle OEB is isosceles triangle (the radii OB and OD \r\n" );
document.write( "are congruent). It has the angle B equal to 60°.\r\n" );
document.write( "It implies that the triangle OEB is equilateral.\r\n" );
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document.write( "Due to similar reasons, the triangle ODA is equilateral, too.   \r\n" );
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document.write( "Thus the triangles OEB and ODA are similar to the triangle \r\n" );
document.write( "ABC with the similarity coefficient .\r\n" );
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document.write( "              Figure 1. \r\n" );
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document.write( "              Figure 2. \r\n" );
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Therefore, the areas of the triangles OEB and ODA are  of the area of the triangle ABC.\r\n" );
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document.write( "It means that the area of the quadrilateral CDOE (which is a rhombus, by the way) is half of the area of the triangle ABC.\r\n" );
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document.write( "Then the area of the curved shape CDE is the area of the rhombus CDOE minus the area of the sector ODE.\r\n" );
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document.write( "In other words,  =  = .\r\n" );
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