Triangle Altitude Length

triallen.html

Exploration

Show that the length, L, of a triangle's altitude (from vertex to side ) is given by

where , and are the lengths of the triangle's sides.

Approach

Construct a triangle in a convenient, yet sufficiently general position.  Then construct the triangle's altitude.  Show that the length of the altitude is given by the expression.  Since the length of each triangle side, , is positive, .

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 a triangle with sides , and .  By default, the triangle's first vertex is located at the origin.

Clear[s1,s2,s3,E1];
T1=Triangle2D[{s1,s2,s3}] /. Sqrt[-E1_/s3^2]->Sqrt[-E1]/s3

The length of the altitude is the distance from the triangle's third vertex to the x-axis.

Lx=Line2D[0,1,0];
altitude=Distance2D[Point2D[T1,3],Lx] /. Sqrt[E1_/s3^2]->Sqrt[E1]/s3