Area of Triangle Bounded by Lines

triarlns.html

Exploration

Show that the area of the triangle bounded by the lines

and x=0

is given by

.

Approach

Create the triangle and compute the area.

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

Create the triangle.

Clear[m1,c1,m2,c2];
t1=Triangle2D[Line2D[m1,-1,c1],
              Line2D[m2,-1,c2],
              Line2D[1,0,0]]

Get the vertex points of the triangle.

{p1,p2,p3}=Map[Point2D[t1,#]&,{1,2,3}]

Compute the area of the triangle using Heron's formula.

Clear[s];
a=Distance2D[p1,p2];
b=Distance2D[p2,p3];
c=Distance2D[p3,p1];
s=(a+b+c)/2;
A1=Sqrt[s(s-a)(s-b)(s-c)] //FullSimplify

Since is positive, the formula simplifies to the desired result.

A1 /. Sqrt[E1_^4/E2_^2]->
      E1^2/Sqrt[E2^2]