document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=198824>Question 198824</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let A(-2,4), B(-7,1) and C(-1,-5) be the vertices of a triangle.<BR>\n" );
document.write( "a. Find an equation whose graph is a line that contains the median from A to the midpoint of line segment BC.<BR>\n" );
document.write( "b. Find the length of the median from A to line segment BC.<BR>\n" );
document.write( "c. Find the length of the altitude from A to line segment BC.<BR>\n" );
document.write( "d. Find the area of the traingle ABC.\r<BR>\n" );
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document.write( "Thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #149323 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=149323\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "<i>Let A(-2,4), B(-7,1) and C(-1,-5) be the vertices of a triangle.</i>\r<BR>\n" );
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document.write( "<i>a. Find an equation whose graph is a line that contains the median from A to the midpoint of line segment BC.</i>\r<BR>\n" );
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document.write( "Use the midpoint formulas to find the midpoint of the segment BC.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_m\ = \frac{x_1 + x_2}{2}\"> and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_m\ = \frac{y_1 + y_2}{2}\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_1,y_1)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_2,y_2)\"> are the coordinates of  points B and C. (doesn't matter which is which)\r<BR>\n" );
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document.write( "Then use the two-point form of the equation of a line to derive the desired equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y - y_1 = \left(\frac{y_1 - y_m}{x_1 - x_m}\right)(x - x_1) \">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_1,y_1)\"> are the coordinates of point A and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_m,y_m)\"> are the coordinates of the midpoint derived in the previous step.\r<BR>\n" );
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document.write( "<i>b. Find the length of the median from A to line segment BC.</i>\r<BR>\n" );
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document.write( "Use the distance formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d = sqrt{(x_1 - x_m)^2 + (y_1 - y_m)^2}\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_1,y_1)\"> are the coordinates of point A and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_m,y_m)\"> are the coordinates of the midpoint derived in the previous step.\r<BR>\n" );
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document.write( "<i>c. Find the length of the altitude from A to line segment BC.</i>\r<BR>\n" );
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document.write( "<b>Step 1.</b>  Derive the equation of the line containing segment BC.  Use the two-point form of the equation of a line:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y - y_1 = \left(\frac{y_1 - y_2}{x_1 - x_2}\right)(x - x_1) \">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_1,y_1)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_2,y_2)\"> are the coordinates of  points B and C.  Put the equation into slope-intercept form (<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y = mx + b\">\r<BR>\n" );
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document.write( "<b>Step 2.</b> Determine the slope of the line that contains the altitude segment, namely a line perpendicular to segment BC passing through A.  First you need the slope of the line containing BC which you can obtain by inspection of the slope-intercept form of the equation of the line containing BC derived in Step 1.\r<BR>\n" );
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document.write( "The slope of the line containing the altitude segment is the negative reciprocal of the slope of the line containing BC because:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  L_1 \perp L_2 \ \ \Leftrightarrow\ \ m_1 = -\frac{1}{m_2} \text{ and } m_1, m_2 \neq 0\">\r<BR>\n" );
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document.write( "<b>Step 3.</b> Derive the equation of the line containing the altitude segment.  Use the point-slope form of the equation of a line:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y - y_1 = m(x - x_1) \">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_1,y_1)\"> are the coordinates of point A and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large m\">  is the slope number calculated in step 2.  Put this equation in slope-intercept form as well.\r<BR>\n" );
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document.write( "<b>Step 4.</b> Solve the system of equations derived in steps 1 and 3 to determine the point of intersection between the altitude and segment BC.  Since both equations are in slope intercept form, you can simply equate the two right-hand sides and solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">, then substitute back into either equation to calculate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">.  This is where this problem gets really ugly -- for example the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the point of intersection is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{-71}{41}\"> -- like a mud fence.  But persevere and you will be rewarded in the end.\r<BR>\n" );
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document.write( "<b>Step 5.</b> Use the distance formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d = sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_1,y_1)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_2,y_2)\"> are the coordinates of  point A and the point of intersection calculated in step 4.\r<BR>\n" );
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document.write( "<i>d. Find the area of the traingle ABC.</i>\r<BR>\n" );
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document.write( "Use the distance formula to calculate the measure of segment BC:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d = sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_1,y_1)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x_2,y_2)\"> are the coordinates of  points B and C.  Call this the measure of the base of the triangle.\r<BR>\n" );
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document.write( "Use the area of a triangle formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A = \frac{bh}{2}\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\"> is the measure of the base and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h\"> is the measure of the altitude calculated in part c of the question.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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