Q : 17      In the triangle ABC with vertices  A(2,3)B(4,-1) and  C(1,2) , find the equation and length of altitude from the vertex A.

Answers (1)
G Gautam harsolia

exercise 10.3
Let suppose foot of perpendicular is (x_1,y_1)
We can say that line passing through point (x_1,y_1) \ and \ A(2,3)  is perpendicular to line passing through point B(4,-1) \ and \ C(1,2)
Now,
Slope of line passing through point B(4,-1) \ and \ C(1,2) is , m' = \frac{2+1}{1-4}= \frac{3}{-3}=-1
And
Slope of line  passing through point (x_1,y_1) \ and \ (2,3) is , m
lines are perpendicular
Therefore,
m = -\frac{1}{m'}= 1
Now,  equation of line passing through point  (2 ,3)  and slope with 1
(y-3)=1(x-2)
x-y+1=0                     -(i)
Now, equation line passing through point B(4,-1) \ and \ C(1,2) is 
(y-2)=-1(x-1)
x+y-3=0
Now, perpendicular distance of (2,3) from the x+y-3=0 is 
d= \left | \frac{1\times2+1\times3-3}{\sqrt{1^2+1^2}} \right |= \left | \frac{2+3-3}{\sqrt{1+1}} \right |= \frac{2}{\sqrt{2}}=\sqrt2                -(ii)

Therefore, equation and length of the line is  x-y+1=0  and  \sqrt2   respectively


 

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