document.write( "Question 874920: PLEASE HELP. SHOW THAT THE LINES $\"+X+-+Y+-+8+=+0+\"$ , $\"+3X+%2B+2Y+%2B+1+=+0+\"$ , $\"+X+-+Y+-+3+=+0+\"$ AND $\"+2X+%2B+3Y+%2B+24+=+0+\"$ ARE THE SIDES OF AN ISOSCELES TRAPEZOID AND FIND ITS AREA. \n" ); document.write( "

Algebra.Com's Answer #528171 by KMST(5328) $\"\"$ $\"About$

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Since $\"system%28red%28x-y-8=0%29%2Cgreen%28x-t-3=0%29%29\"$ has no solution,
\n" ); document.write( "the lines $\"red%28x-y-8=0%29\"$ and $\"green%28x-y-3=0%29\"$ are parallel.
\n" ); document.write( "What's more, the slope of the lines is $\"1\"$ ,
\n" ); document.write( "since $\"red%28x-y-8=0%29\"$ --> $\"red%28y=x-8%29\"$
\n" ); document.write( "and $\"green%28x-y-3=0%29\"$ --> $\"green%28y=x-3%29\"$ .
\n" ); document.write( "The slope of the lines is $\"1\"$ .
\n" ); document.write( "Since $\"tan%2845%5Eo%29=1\"$ , the lines make a $\"45%5Eo\"$ with the positive x-axis.
\n" ); document.write( "The bases are part of those parallel lines, which make a $\"45%5Eo\"$ with the positive x-axis.
\n" ); document.write( "A perpendicular to those lines from the y-intercept of $\"green%28y=x-3%29\"$ at (0,-3} forms an isosceles right triangle with $\"red%28y=x-8%29\"$ and the y-axis, with a hypotenuse of $\"8-3=5\"$ . So the length of the legs of that triangle is
\n" ); document.write( " $\"5sin%2845%5Eo%29=5sqrt%282%29%2F2\"$ , and that is the distance between the lines and the height of the trapezoid.
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\n" ); document.write( "The vertices of the trapezoid can be calculated as the intersections of the lines.
\n" ); document.write( " $\"system%28red%28y=x-8%29%2C3x%2B2y%2B1=0%29\"$ --> $\"system%28x=3%2Cy=-5%29\"$
\n" ); document.write( " $\"system%28red%28y=x-8%29%2C2x%2B3y%2B24=0%29\"$ --> $\"system%28x=0%2Cy=-8%29\"$
\n" ); document.write( " $\"system%28green%28y=x-3%29%2C3x%2B2y%2B1=0%29\"$ --> $\"system%28x=1%2Cy=-2%29\"$
\n" ); document.write( " $\"system%28green%28y=x-3%29%2C2x%2B3y%2B24=0%29\"$ --> $\"system%28x=-3%2Cy=-6%29\"$
\n" ); document.write( "The vertices are (3,-5) and (0,-8) for the base on $\"red%28y=x-8%29\"$
\n" ); document.write( "The vertices are (1,-2) and (-3,-6) for the base on $\"green%28y=x-3%29\"$
\n" ); document.write( "The length of the bases is the distance between their vertices, so those lengths are
\n" ); document.write( " $\"sqrt%28%283-0%29%5E2%2B%28-5%2B8%29%5E2%29=sqrt%289%2B9%29=sqrt%282%2A9%29=3sqrt%282%29\"$
\n" ); document.write( "and $\"sqrt%28%281%2B3%29%5E2%2B%28-2%2B6%29%5E2%29=sqrt%2816%2B16%29=sqrt%282%2A16%29=4sqrt%282%29\"$
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\n" ); document.write( "The area of a trapezoid (isosceles or not) can be calculated as
\n" ); document.write( " $\"In+this+case+the+area+is%0D%0A%281%2F2%29%28base%5B1%5D%2Bbase%5B2%5D%29%28height%29\"$
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and in close-up

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\n" ); document.write( "To prove that it is an isosceles trapezoid, we could easily prove that the non-parallel sides (the legs) are congruent.
\n" ); document.write( "The length of one of those sides is the distance between vertices on $\"2x%2B3y%2B24=0\"$ , (0,-8) and(-3,-6).
\n" ); document.write( "That distance is
\n" ); document.write( " $\"sqrt%28%280%2B3%29%5E2%2B%28-8%2B6%29%5E2%29=sqrt%289%2B4%29=sqrt%2813%29\"$ .
\n" ); document.write( "The length of the other leg is the distance between vertices on
\n" ); document.write( " $\"3x%2B2y%2B1=0%29\"$ , (1,-2) and (3,-5).
\n" ); document.write( "That distance is
\n" ); document.write( " $\"sqrt%28%281-3%29%5E2%2B%28-2%2B5%29%5E2%29=sqrt%284%2B9%29=sqrt%2813%29\"$ . \n" ); document.write( "

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