Dot Product, what's wrong with my method?

[rocomath]

Find the three angles of the triangle with given vectors.

A(1,0)
B(3,6)
C(-1,4)

I found that AB & BC are congruent, so this ends up being an Isosceles triangle and the only angle I need to find is B.

BC=<3+1,6-4>=<4,2>
AB=<3-1,6-0>=<2,6>

$$\angle B=\cos^{-1}{\frac{20}{\sqrt{20 \cdot 40}}=45^o$$

So $$\angle A = \angle C = \frac{180-45}{2}=67.5^o$$ but this isn't correct, so what am I doing wrong?

It's supposed to be a 45, 45, 90 triangle.

 

[Dick]

What do you mean AB and BC are congruent? That they have the same length? I don't think so. You used that their lengths are sqrt(20) and sqrt(40). AC and BC are the two equal sides.

 

[dynamicsolo]

Angle A is not congruent to angle C; it is angles B and C that are congruent. Check the dot product of vector AC with vector AB. (This is indeed a right isosceles triangle.) Making a picture of the situation is often helpful in keeping straight which vectors you want to look at.

 

[rocomath]

What do you mean AB and BC are congruent? That they have the same length? I don't think so. You used that their lengths are sqrt(20) and sqrt(40). AC and BC are the two equal sides.

Angle A is not congruent to angle C; it is angles B and C that are congruent. Check the dot product of vector AC with vector AB. (This is indeed a right isosceles triangle.) Making a picture of the situation is often helpful in keeping straight which vectors you want to look at.

Thanks! I found my mistake in mislabeling my congruent sides :-[