The third vertex of a triangle

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Question from nikos, a student:

Hi, I would really appreciate any help in this:

"Suppose you have a triangle (any triangle, no hypotheses allowed), let's call it ABC.
You know the coordinates of the two vertexes, A(x1,y1), B(x2,y2) and the area of the triangle, let's call it D.
Can you find the third vertex of this triangle based on these facts?"

Hi Nikos,

There are many possible choices for the third vertex. Suppose that the line segment AB is the base of the triangle and the distance from A to B is b. Suppose the given area of the triangle is T square units then

T = 1/2 b × h square units

where h is the height of the triangle. You know b and T so any point C that is h units from the line through AB forms a triangle ABC with area T square units. Here is a construction technique which will produce two such points C.

Use the distance formula to find b, the distance from A to B. Find the slope m of the line through A and B. Any line perpendicular to the line through a and B has slope -1/m. Write the equation of the line through A with slope -1/m. Let C(x₃, y₃) be a point on this line and solve for the coordinates of C so that the distance from A to C is h = 2T/b.

If you have difficulty finding the coordinates of C tell us what you have done and where you are stuck and we will try to help.

Harley

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.