Find unknown vertex of triangle given area and other 2 vertices

Question

I need to find the coordinates of the 3rd vertex of a triangle given that I know the other 2 vertices and the area. The triangle is not guaranteed to be of any particular type (right, isosceles, acute, etc..).

Known vertex 1 will have coords (x1,y1)

known vertex 2 will have coords (x2,y2)

unknown vertex 3 will have coords (?, y2) or (x2, ?), in other words one part of the coordinate is the same as that in vertex 2.

Thanks

Answer 1

You're given one side, so you can calculate the length of that side. Next, you use the formula for area of a triangle to determine the height of the triangle. Finally, you set up the expression to calculate the distance from a point to a line using the known coordinate and the height you just calculated.

Answer 2

Given two vertices and the area, there are an infinite number of triangles with these. Draw the line through the two vertices and choose any point that makes the resulting triangle have the desired area. Draw the line through that point parallel to the line through the two vertices.

Then a triangle with any point on that line and the original vertices has the same area.

tl;dr A = bh/2.

Answer 3

Suppose that vertex 3 has co-ordinates $(x_2, Y)$. Then vertex 2 and vertex 3 are on the same vertical line $x = X_2$.

The length of altitude from vertex 1 is given by $|X_2 – X_1|$.

The above, when combined with the given area, will give you L, the length of base (formed by vertex 2 and vertex 3).

Then Y is given by either $Y_2 + L$ or $Y_2 – L$.

The case for vertex 3 $= (?, y_2)$ can be treated in a similar manner.