If I have the triangle ABC, that A(3,4) B(1,2) C(6,1)
How can I find the value of angle A?
Please, thank you
Hello,
from the title of your post I assume that you have to use vectors (If you don't want to use vectors you can solve this problem by using Cosine rule)
1. Calculation of the vectors:
$\displaystyle \overrightarrow{AB}=(1,2)-(3,4)=(-2,-2)$
$\displaystyle \overrightarrow{AC}=(6,1)-(3,4)=(3,-3)$
2. The angle $\displaystyle \alpha$ between 2 vectors $\displaystyle \vec u$ and $\displaystyle \vec v$ is calculated by:
$\displaystyle \cos(\alpha)=\frac{\vec u \cdot \vec v}{|\vec u| \cdot |\vec v|}$
Plug in the vectors you know into this formula:
$\displaystyle \cos(A)=\frac{(-2,-2) \cdot (3,-3)}{\sqrt{8} \cdot \sqrt{18}}=\frac{0}{12}=0$
Therefore A = 90° : You are dealing with a right triangle.
3. Hint: Draw a sketch of this triangle to prove the calculations.
EB
Calculate the squares of the lengths of the sides:
AB^2=(3-1)^2+(4-2)^2=8
AC^2=(3-6)^2+(4-1)^2=18
BC^2=(1-6)^2+(2-1)^2=26.
So AB^2+AC^2=BC^2. Therefore by the converse of Pythagoras's theorem
ABC is a right triangle with BC as hypotenuse, and the angle at A is a right
angle.
RonL