5/12/13 — Happy Right Triangle Day!

Happy Right Triangle Day!  Today we celebrate a favorite geometric object:  the 5-12-13 right triangle.

Of course, the sides of this triangle satisfy the Pythagorean Theorem

$5^2 + 12^2 = 13^2$

but one reason I like this particular right triangle so much is the role it plays in another favorite triangle.  The 5-12-13 triangle fits together perfectly with the 9-12-15 right triangle

to make the 13-14-15 triangle!

The 13-14-15 triangle is special in its own right:  it is a Heronian triangle, a triangle with rational side lengths and rational area.  In fact, this triangle has  integer side lengths and integer area, making it especially interesting!

Happy Right Triangle Day!  Be sure to marvel at some perpendicularity today.

Categories: AppreciationGeometry

Simon Gregg · May 12, 2013 at 4:48 pm

All right, Mr Honner! Happy Right Triangle Day!

I liked this lots. Looking around I found another way of splitting up the 13-14-15 triangle which I like too.

Simon Gregg · May 19, 2013 at 10:46 am

(I’ve now added a little incircle goodness to that post.)

You wrote 13^3.

MrHonner · May 12, 2013 at 5:55 pm

Thanks for the edit! Fixed.

Nat · May 12, 2013 at 7:01 pm

Happy Right Triangle Day! I think you should also check out the work of Michelle Desmarais. She has done incredible work pertaining to primitive Pythagorean Triples and possible extensions to the Pythagorean theorem.

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