document.write( "Question 1107299: The sides of a triangle lie along the straight line with equations y=1; x+y=6 and y-3x+2=0.
\n" ); document.write( "(a) Find the equation of the altitudes. (An altitude is a line through a vertex perpendicular to the opposite)
\n" ); document.write( "(b) Show that the altitudes are con-current and find the coordinates of the point where they meet (the point is called the orthocentre of the triangle). \n" ); document.write( "

Algebra.Com's Answer #722366 by KMST(5328) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "a) The vertices are
\n" ); document.write( " $\"A%285%2C1%29\"$ , given by
\n" ); document.write( " $\"system%28y=1%2Cx%2By=6%29\"$ --> $\"system%28y=1%2Cx%2B1=6%29\"$ --> $\"system%28y=1%2Cx=5%29\"$ ,
\n" ); document.write( " $\"B%281%2C1%29\"$ , given by
\n" ); document.write( " $\"system%28y=1%2Cy-3x%2B2=0%29\"$ --> $\"system%28y=1%2C1-3x%2B2=0%29\"$ --> $\"system%28y=1%2C3-3x=0%29\"$ --> $\"system%28y=1%2Cx=1%29\"$ , and
\n" ); document.write( " $\"C%282%2C4%29\"$ , given by
\n" ); document.write( " $\"system%28x%2By=6%2Cy-3x%2B2=0%29\"$ --> $\"system%28x%2By=6%2C3x-y=2%29\"$ --> $\"system%28x%2By=6%2C4x=8%29\"$ --> $\"system%28x%2By=6%2Cx=2%29\"$ --> $\"system%282%2By=6%2Cx=2%29\"$ --> $\"system%28y=4%2Cx=2%29\"$ .
\n" ); document.write( "Side $\"AB\"$ is the horizontal line $\"y=1\"$ .
\n" ); document.write( "The altitude from $\"C%282%2C4%29\"$ to $\"AB\"$ , perpendicular to $\"AB\"$
\n" ); document.write( "is the vertical line $\"highlight%28x=2%29\"$
\n" ); document.write( "Side $\"AC\"$ is the line $\"x%2By=6\"$ <--> $\"y=6-x\"$ , with slope $\"-1\"$ .
\n" ); document.write( "The altitude from $\"B%281%2C1%29\"$ to $\"AC\"$ , perpendicular to $\"AC\"$
\n" ); document.write( "having slope $\"1\"$ , is
\n" ); document.write( " $\"y-1=1%28x-1%29\"$ --> $\"y=x-1%2B1\"$ --> $\"highlight%28y=x%29\"$ .
\n" ); document.write( "Side $\"BC\"$ is the line $\"y-3x%2B2=0\"$ <--> $\"y=3x-2\"$ , with slope $\"3\"$ .
\n" ); document.write( "The altitude from $\"A%285%2C1%29\"$ to $\"BC\"$ , perpendicular to $\"BC\"$
\n" ); document.write( "having slope $\"-1%2F3\"$ , is
\n" ); document.write( " $\"y-1=%28-1%2F3%29%28x-5%29\"$ --> $\"y-1=%28-1%2F3%29x%2B5%2F3\"$ --> $\"y=%28-1%2F3%29x%2B5%2F3%2B1\"$ --> $\"highlight%28y=%28-1%2F3%29x%2B8%2F3%29\"$ .
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\n" ); document.write( "\n" ); document.write( "The intersection of $\"x=2\"$ and $\"y=x\"$ is given by $\"P%282%2C2%29\"$
\n" ); document.write( " $\"system%28x=2%2Cy=x%29\"$ --> $\"system%28x=2%2Cy=2%29\"$ .
\n" ); document.write( "The intersection of $\"x=2\"$ and $\"y=%28-1%2F3%29%2B8%2F3\"$ is the solution to
\n" ); document.write( " $\"system%28x=2%2Cy=%28-1%2F3%29x%2B8%2F3%29\"$ --> $\"system%28x=2%2Cy=-2%2F3%2B8%2F3%29\"$ --> $\"system%28x=2%2Cy=6%2F3%29\"$ --> $\"system%28x=2%2Cy=2%29\"$ ,
\n" ); document.write( "also point $\"P%282%2C2%29\"$ .
\n" ); document.write( "The intersection of $\"y=x\"$ and $\"y=%28-1%2F3%29%2B8%2F3\"$ is the solution to
\n" ); document.write( " $\"system%28y=x%2Cy=%28-1%2F3%29x%2B8%2F3%29\"$ --> $\"system%28y=x%2Cx=%28-1%2F3%29x%2B8%2F3%29\"$ --> $\"system%28y=x%2Cx%2B%281%2F3%29x=8%2F3%29\"$ --> $\"system%28y=x%2C%284%2F3%29x=8%2F3%29\"$ --> $\"system%28y=x%2Cx=2%29\"$ --> $\"system%28y=2%2Cx=2%29\"$ ,
\n" ); document.write( "also point $\"P%282%2C2%29\"$ . \n" ); document.write( "

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