So the thing will be to find the equations of the lines representing the altitudes.
For example, we'll look for the altitudes coming from R (to ST) and S (to RT).
Equations of the lines ST and RT.
We know that to a line of equation ax+by+c=0 corresponds a direction vector (-b,a) and an orthogonal vector (a,b).
So here, -b=3 and a=-7.
Therefore, according to the red part, the equation of line ST is : -7x-3y+c=0.
But we're interested in the orthogonal line to ST, passing through R.
Its vector director is, according to the red part above : (-7,-3).
So here again, to find out the equation of this line, we use the red part, with -b=-7 (b=7) and a=-3
The equation of a line orthogonal to ST is : -3x+7y+c=0.
i do these steps because I can never remember the equations...
We want this line passing through R(-3,2).
By substituting in the equation :
Therefore, the equation of the altitude coming from R to ST is :
Now, for RT, we work the same way :
Equation of line RT : (thanks to the red part)
Vector director of a line orthogonal to RT : (thanks to the blue part)
Equation of a line orthogonal to RT : (thanks to the red part)
We know the altitude to RT goes through S(4,5).
By substituting :
Therefore, the equation of the altitude coming from S to RT is :
H is on these two altitudes.
So solve the system :
If there are any mistakes, don't hesitate ! I'm likely to have made typos, but the reasoning may be correct.