# Intersection point and angle between the extended hypotenuses of two right-angled triangles in the plane

The end points of two line segments (the hypotenuses of red and blue right-angled triangles below) are given below with their coordinates marked separately on the $x$- and $y$-axes. These points are

• red triangle: $(0,1.13684)$ and $(1,1.13682)$
• blue triangle: $(0,1.1357)$ and $(1,1.1366)$.

Can you help to find the point of intersection $(X,Y)$ if these line segments were extended, and the acute angle $a$ between them? And please also show the working?

$$m_1=\frac{1.13682-1.13684}{1-0}=-0.00002$$
$$m_2=\frac{1.1366-1.1357}{1-0}=0.0009$$
The angle of inclination of a line is the arctangent of the slope, and one of the angles between two lines is the difference of their angles of inclination. Therefore, an angle between your two lines is $$\arctan m_2-\arctan m_1$$ $$=\arctan (0.0009)-\arctan (-0.00002)$$ $$=\arctan (0.0009)+\arctan (0.00002)$$ $$\approx 0.000919999757 \text{ radians}$$ $$\approx 0.052712103229°$$