document.write( "Question 317859: Are the points A(6,4),B(8,6),C(4,8) and D(2,4) vertices of a square? Find its area \n" ); document.write( "

Algebra.Com's Answer #227616 by Edwin McCravy(20056) $\"\"$ $\"About$

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Are the points A(6,4),B(8,6),C(4,8) and D(2,4) vertices of a square? Find its area.
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document.write( "No way is that a square!  Maybe you meant A(6,2) and not A(6,4).  I'll consider\r\n" );
document.write( "both cases in the event that's what you meant.\r\n" );
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document.write( "I will go ahead and find the area of the (non-square) polygon whose vertices\r\n" );
document.write( "are A(6,4), B(8,6), C(4,8), and D(2,4), then I'll look at the case where\r\n" );
document.write( "you might have meant A(6,2) and not A(6,4)\r\n" );
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document.write( "Since the polygon is convex (doesn't \"sink in\" anywhere). we use the\r\n" );
document.write( "determinant formula, whose rows are the coordinates in counter-clockwise\r\n" );
document.write( "order, with the first row repeated at the bottom:\r\n" );
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document.write( "To expand it add the sum of the products of the diagonals going down\r\n" );
document.write( "to the right and subtract the sum of the products of the diagonals\r\n" );
document.write( "going up to the right:\r\n" );
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document.write( "If you meant for point A to be A(6,2) and not (6.4) and just mistyped,\r\n" );
document.write( "then the figure would have been a square.\r\n" );
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document.write( "line(2,4,6,4), line(6,4,8,6), line(8,6,4,8), line(4,8,2,4) )}}} \r\n" );
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document.write( "We can show that is a square by drawing in some (green) right triangles,\r\n" );
document.write( "forming a little square in the middle:\r\n" );
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document.write( "Those 4 green right triangles have one leg 2 units long,\r\n" );
document.write( "and the other leg 4 units long. They are all congruent\r\n" );
document.write( "and so their hypotenuses all have the same length. That\r\n" );
document.write( "shows the quadrilateral is at least a rhombus.  Each internal\r\n" );
document.write( "angle of ABCD is the sum of two complementary angles (acute\r\n" );
document.write( "angles of a right triangle).  I'll let you fill in the details\r\n" );
document.write( "as to why the internal angles of rhombus ABCD are 90° each.\r\n" );
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document.write( "Each of the four green right triangles have an area of \r\n" );
document.write( ".  So four of them makes 16 square units.\r\n" );
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document.write( "The little square in the middle is 2×2 so its area is 4, and so the area\r\n" );
document.write( "of the square is 16+4 or 20 square units.\r\n" );
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document.write( "We could also have used the determinant method which we used to get the \r\n" );
document.write( "first area above which wasn't a square\r\n" );
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document.write( "Edwin

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