Altitude of a Triangle

trialt.html

Exploration

The altitude from vertex A of ΔABC is a line segment from A perpendicular to side BC (or the extension of BC). Show that the equation of the line containing the altitude from A is

where the coordinates of the vertices are , and .

Approach

Construct the altitude and show that the line containing it is the line given.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Construct the line BC.

Clear[x2,y2,x3,y3];
BC=Line2D[{x2,y2},{x3,y3}];

Construct the altitude from A perpendicular to BC.

Clear[x1,y1];
alt=Line2D[Point2D[x1,y1],BC]

Convert the line to an equation.

Clear[x,y];
Equation2D[alt,{x,y}]

Discussion

This defines a new function that constructs all the lines underlying the altitudes of a triangle.

Altitudes2D[Triangle2D[{x1_,y1_},{x2_,y2_},{x3_,y3_}]]:=
   {Altitude$2D[{x1,y1},{x2,y2},{x3,y3}],
    Altitude$2D[{x2,y2},{x3,y3},{x1,y1}],
    Altitude$2D[{x3,y3},{x1,y1},{x2,y2}]};
Altitude$2D[{x1_,y1_},{x2_,y2_},{x3_,y3_}]:=
   Line2D[x3-x2,y3-y2,-x1(x3-x2)-y1(y3-y2)];

This is the plot of a numerical example.

T1=Triangle2D[{-1,-2},{-2,3},{4,0}];
Sketch2D[{T1,Altitudes2D[T1],
          Map[Point2D,List @@ T1]}]

Graphics saved as "trialt01.eps".