Algebra.Com's Answer #743070 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Please help me solve this:\r
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document.write( "\"AB has endpoints A(-5,0) and B(4,3). CD has endpoints C(-3,9) and D(1,-3). The equations of the lines
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document.write( "containing AB and CD are x - 3y = -5 and 3x + y = 0, respectively.\"\r
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document.write( "a. How could you quickly check that these equations are correct?\r\n" );
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document.write( " The simplest way to check that AB lies on the straight line x - 3y = -5 is to check that each endpoint A and B lies on this line.\r\n" );
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document.write( " To check it for AB, simply substitute the coordinates of A x= -5 and y= 0 into the equation:\r\n" );
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document.write( " -5 -3*0 = - 5 (! correct !), \r\n" );
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document.write( " and then substitute the coordinates of B x= 4 and y= 3 into the equation:\r\n" );
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document.write( " 4 -3*3 = - 5 (! correct !).\r\n" );
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document.write( " Thus AB is checked. Now you check for CD by the same way.\r\n" );
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document.write( "b. Verify that the lines are perpendicular.\r\n" );
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document.write( " The line x - 3y = -5 has the slope  .\r\n" );
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document.write( " The line 3x + y = 0 has the slope -3.\r\n" );
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document.write( " The numbers and -3 are opposite reciprocal.\r\n" );
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document.write( " It means that the two lines are perpendicular.\r\n" );
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document.write( "c. Find the point of intersection of AB and CD by solving the system of equations.\r\n" );
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document.write( " x - 3y = -5. (1)\r\n" );
document.write( " 3x + y = 0. (2)\r\n" );
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document.write( " From eq(1) express x = -5 + 3y and substitute it into eq(2). You will get\r\n" );
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document.write( " 3*(-5+3y) + y = 0,\r\n" );
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document.write( " -15 + 9y + y = 0\r\n" );
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document.write( " 10y = 15 ====> y = 15/10 = 1.5.\r\n" );
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document.write( " Then x = -5 + 3y = -5 +3*1.5 = -5 + 4.5 = -0.5.\r\n" );
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document.write( " Thus the intersection point is (x,y) = (-0.5,1.5)\r\n" );
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document.write( "d. Find the midpoints of AB and CD. Compare your results with Part c.\r\n" );
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document.write( " To find midpoint of AB, take x-coordinates of A and B, add them and divide the sum by 2 \r\n" );
document.write( " (so you will get the average of x-coordinates of endpoints A and B).\r\n" );
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document.write( " By doing it, you will get x-coordinate of the midpoint AB as  = -0.5.\r\n" );
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document.write( " Now make the same with y-coordinates of A and B: you will get y-coordinate of the midpoint AB as  = 1.5.\r\n" );
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document.write( " Thus the midpoint of AB is (x,y) = (-0.5,1.5).\r\n" );
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document.write( " To determine the midpoint of CD, do the same with the points C and D.\r\n" );
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document.write( " Notice that is is THE SAME point as the intersection point of AB and CD, which we determined in Part c). \r\n" );
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document.write( "e. What kind of quadrilateral is ACBD? Explain your reasoning.\r\n" );
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document.write( " The quadrilateral ABCD has perpendicular diagonals AB and CD; and the diagonals intersetion is the midpoint of each diagonal.\r\n" );
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document.write( " Hence the quadrilateral ABCD is a rhombus.\r\n" );
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document.write( "COMPLETED.\r
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