Exploring Analyic Geometry with Mathematica®

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Hypotenuse Midpoint Distance

tridist.html

Exploration

Prove that the midpoint of the hypotenuse of a right triangle is equidistant from the vertices.

Approach

Without loss of generality, create a triangle in a convenient position with the right angle vertex at the origin and the other two vertices at (a,0) and (0,b). Create the midpoint of the hypotenuse and then examine the distance from the midpoint to each of the vertices.

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 points defining the triangle's vertices.

Clear[a,b];
p1=Point2D[0,0];
p2=Point2D[a,0];
p3=Point2D[0,b];

Construct the midpoint of the hypotenuse.

P=Point2D[p2,p3]

"tridist_1.gif"

The distances from the midpoint to the vertices are equal by inspection.

Map[Distance2D[P,#]&,{p1,p2,p3}]

"tridist_2.gif"

Discussion

This is the plot of a numerical example.

Sketch2D[{p1,p2,p3,P,
          Segment2D[p1,P],
          Segment2D[p2,p3]} /.
         {a->3, b->2}]      

"tridist_3.gif"

Graphics saved as "tridis01.eps".


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