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Triangle Altitude Length

triallen.html

Exploration

Show that the length, L, of a triangle's altitude (from vertex "triallen_1.gif" to side "triallen_2.gif") is given by

"triallen_3.gif"

where "triallen_4.gif", "triallen_5.gif" and "triallen_6.gif" 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, "triallen_7.gif", is positive, "triallen_8.gif".

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 "triallen_9.gif", "triallen_10.gif" and "triallen_11.gif".  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

"triallen_12.gif"

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

"triallen_13.gif"


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
www.Descarta2D.com