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The points (3, 7), (6, 2) and (2,k) are the vertices of a triangle. For how many real values of k is the triangle a right triangle?

 Dec 29, 2020
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(3,7)  (6, 2)   ( 2, k)

 

We need to find the slope  between   (3,7)   and (6,2)

 

(7 - 2)   / ( 3  - 6)  =  -5/3

 

Now  we need to find  two  lines with  reciprocal slopes  ( perpendicular to AB)  that pass through (3,7)  and (6,2)

 

So...one line has the equation

 

y = (3/5) ( x - 3)  + 7

y = (3/5)x - 9/5 + 35/5

y = (3/5)x + 26/5

 

So one value of  k  is

 

k = (3/5)(2) + 26/5

k = 6/5 + 26/5

k = 32/5

 

And  the  other line has the equation

y = (3/5)( x - 6)  + 2

y = (3/5)x - 18/5 + 10/5

y = (3/5)x -8/5

 

So  the other value of  k is

 

k  = (3/5)(2)  - 8/5

k = 6/5 - 8/5

k  = -2/5

 

 

cool cool cool

 Dec 29, 2020

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