Area of Triangle Configurations

triarea.html

Exploration

Graphics saved as "triare01.eps".

For the triangle illustrated in the figure, show that the area, , associated with the AAS (angle-angle-side) configuration whose parameters are , and is given by

.

Show that the area, , associated with the ASA (angle-side-angle) configuration whose parameters are , and is given by

.

Show that the area, , associated with the SAS (side-angle-side) configuration whose parameters are , and is given by

.

Approach

Construct the triangle associated with each configuration, and then compute the expression for the area and simplify.

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 triangle for the AAS configuration.

Clear[a1,a2,a3,s1,s2,s3];
T1=(Triangle2D[{{s1,Null,Null},{a1,a2,Null}}] //FullSimplify) //.
      Sqrt[s1^2*Sin[a2]^2]->s1*Sin[a2]

Compute the area for the AAS configuration and simplify.  The sine function is introduced to prevent simplification back to the cosecant form.

A1=Area2D[T1] //. Csc[a1]->1/sine[a1]

Construct the triangle for the ASA configuration.

Clear[a1,a2,a3,s1,s2,s3];
T2=(Triangle2D[{{Null,Null,s3},{a1,a2,Null}}] //FullSimplify) //.
      Sqrt[s3^2*Csc[a1+a2]^2*Sin[a1]^2*Sin[a2]^2]->
      s3*Csc[a1+a2]*Sin[a1]*Sin[s2]

Compute the area for the ASA configuration and simplify.  The sine function is transformed to lower case to prevent simplification back to the cosecant form.

A2=Area2D[T2] //. Csc[a1+a2]->1/sine[a1+a2]

Construct the triangle for the SAS configuration.

Clear[a1,a2,a3,s1,s2,s3];
T3=(Triangle2D[{{s1,Null,s3},{Null,a2,Null}}] //FullSimplify) //.
      Sqrt[a1^2*Sin[a2]^2]->s1*Sin[a2]

Compute the area for the SAS configuration and simplify.

A3=Area2D[T3] //. Sqrt[s1^2*Sin[a2]^2]->s1*Sin[a2]