given the points A(0,0), B(3,1) and C (1,4)

Question

what is the measure of angle ABC
I plotted it on a gragh and got
1.46 degress. I am unsure if I am doing this question correctly. Or am I misreading it
thanks in advance .

Answer 1

A simple sketch will show you that that answer cannot possibly be right

Extend CB to cut the x-axis at D
(using the basic concept that if the slope of a line is m and the line forms an angle of Ø with the x-axis
then tanØ = m)

slope of AB = 1/3
so tan BAD = 1/3
BAD = 18.43°
slope of CB = -3/2 , angle is obtuse
tan (angle of line with x-axis) = 123.69°
so angle BDA = 56.304°
and angle ABC is the exterior angle
and angle ABC = angle BAD + angle BDA
= 18.43° + 56.304° = 74.74°

or

If you know vectors
vector CB = (-2,3)
vector BA = (-3,-1)

v(CB) ∙ v(BA) = |CB||BA)cosØ, where Ø is our angle
6 - 3 = √13 √10 cos‚
cosØ = 3/√130 = .26311..
Ø = 74.74° as above.