Find the value of the angle X

Question

I need to solve this problem, but I don't know where should I begin, is there someone who can help me solving this.

The text says: Find the value of the angle painted with red (angle X)

Thanks in advance.

Answer 1

Let $\alpha$ be the angle between the base and the pink line and $\beta$ be the angle between dark blue line and the base. Then, $x = \alpha -\beta$. We already know $\tan\alpha = 2, \tan\beta = 1/3$. Thus, $$\tan(\alpha-\beta) = {\tan\alpha -\tan\beta\over1+\tan\alpha\tan\beta} = {5/3\over 5/3} = 1$$ Thus, $x = 45^o$.

Answer 2

An alternative method using analytical geometry.

The equations of the blue and pink straight lines with respect to natural axes are resp.

$$\begin{cases}y&=&2x-2\\y&=&\frac13x\end{cases}$$

Their intersection point is found to be: $(x,y)=(1.2,0.4).$

It suffices now to consider vectors $\|\vec{V_1}\|=\binom{1.8}{0.6}$ and $\|\vec{V_2}\|=\binom{0.8}{1.6}$ (do you see them on the figure ?) to deduce that :

$$\cos(\theta)=\dfrac{\vec{V_1}.\vec{V_2}}{\|\vec{V_1}\|\|\vec{V_2}\|}=(12/5)/( (12/5)\sqrt{2})=\dfrac{1}{\sqrt{2}}.$$

Thus $\theta=45°.$

Answer 3

It's $\arctan2-\arctan\frac{1}{3}=45^{\circ}$.