SOLUTION: A triangle is defined by the three points: A=(5,2) B=(7,5) C=(8,4). Determine all three angles in the triangle (in radians). θa = θb = θc =

Algebra ->  Vectors -> SOLUTION: A triangle is defined by the three points: A=(5,2) B=(7,5) C=(8,4). Determine all three angles in the triangle (in radians). θa = θb = θc =       Log On


   

Question 968332: A triangle is defined by the three points:
A=(5,2)
B=(7,5)
C=(8,4).
Determine all three angles in the triangle (in radians).
θa =
θb =
θc =
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
A=(5,2)
B=(7,5)
C=(8,4). 


This could be done with lengths of sides and the law of cosines but it
is more easily done with angles between lines using slopes.

All three of those angles appear to be acute (less than 90°)

The acute angle between two lines is given by 

tan%28theta%29%22%22=%22%22abs%28%28m%5B1%5D-m%5B2%5D%29%2F%281%2Bm%5B1%5Dm%5B2%5D%29%29 

∠A is the angle between AB and AC.

We find the slope of AB and call it m1

m%5B1%5D%22%22=%22%22%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29

m%5B1%5D%22%22=%22%22%285-2%29%2F%287-5%29

m%5B1%5D%22%22=%22%223%2F2

We find the slope of AC and call it m2

m%5B2%5D%22%22=%22%22%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29

m%5B2%5D%22%22=%22%22%284-2%29%2F%288-5%29

m%5B2%5D%22%22=%22%222%2F3

Substitute these in:

tan%28A%29%22%22=%22%22abs%28%28m%5B1%5D-m%5B2%5D%29%2F%281%2Bm%5B1%5Dm%5B2%5D%29%29 

tan%28A%29%22%22=%22%22abs%28%283%2F2-2%2F3%29%2F%281%2Bexpr%283%2F2%29expr%282%2F3%29%29%29 

tan%28A%29%22%22=%22%22abs%28%283%2F2-2%2F3%29%2F%281%2B1%29%29

tan%28A%29%22%22=%22%22abs%28%283%2F2-2%2F3%29%2F2%29

Multiply top and bottom by LCD of 6

tan%28A%29%22%22=%22%22abs%28%289-4%29%2F12%29

tan%28A%29%22%22=%22%22abs%285%2F12%29

tan%28A%29%22%22=%22%225%2F12

A%22%22=%22%22%2222.61986595%B0%22

Notice that instead of calling the angle ∠A, here 
it is called θA. 

Find the other two angles the same way.

Edwin