Question 1064074
Segment BD is perpendicular to segment CA.  What is the exact intersection point of lines BD and CA?  How far is point D from point B?



line CA
{{{y-3=((4-3)(2-8))(x-8)}}}
{{{y-3=-(1/6)(x-8)}}}
slope is  {{{-1/6}}}.



line BD
Need slope to be  6.
{{{y-5=6(x-7)}}}



Now, again, what is the intersection of line CA and BD?
{{{system(y-3=-(1/6)(x-8),y-5=6(x-7))}}}

{{{system(6y-18=-x+8,y=6(x-7)+5)}}}

{{{system(x+6y=26,y=6x-42+5)}}}

{{{system(x+6y=26,y=6x-37)}}}

{{{system(x+6y=26,6x-y=37)}}}

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{{{x+6(6x-37)=26}}}

{{{x+36x-6*37=26}}}

{{{37x=26+6*37}}}

{{{highlight(x=248/37)}}}

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{{{y=6(248/37)-37}}}

{{{y=6*248/37-37^2/37}}}

{{{y=(6*248-37^2)/37}}}

{{{highlight(y=119/37)}}}

Point D is at {{{system(x=248/37,y=119/37)}}}



What is the distance between D and B?   This is the altitude for the base AC.


Area for the triangle,  {{{(1/2)*AC*BD}}}