SOLUTION: In ∆ ABC , a = 7 , b= 11 , c = 8 .Calculate the projection of AC on AB.

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Question 1066785: In ∆ ABC , a = 7 , b= 11 , c = 8 .Calculate the projection of AC on AB.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
a%5E2%2Bc%5E2=8%5E2%2B7%5E2=64%2B49=113%3C121=11%5E2=b%5E2 ,
so ABC is an obtuse triangle,
with the obtuse angle opposite the longest side, b=11 ,
at vertex B .
Triangle ABC , with its height h=PC , looks like this:
The projection of segment AC on line AB is segment AP .
Let's call the length of that projection x=AP , and figure out how ti find x .
Apply the Pythagorean theorem to right triangles APC and BPC .
For BPC :
%28x-8%29%5E2%2Bh%5E2=7%5E2<--->x%5E2-16x%2B8%5E2%2Bh%5E2=49<--->x%5E2-16x%2B64%2Bh%5E2=49<--->x%5E2-16x%2Bh%5E2=49-64<--->x%5E2-16x%2Bh%5E2=-15 .
For APC :
x%5E2%2Bh%5E2=11%5E2<--->x%5E2%2Bh%5E2=121 .
Subtracting x%5E2-16x%2Bh%5E2=-15 from x%5E2%2Bh%5E2=121 , we get
16x=121%2B15--->16x=136--->x=136%2F16--->highlight%28x=17%2F2=8.5%29 .