Advanced High-School Mathematics

(Tina Meador) #1

SECTION 1.2 Triangle Geometry 3


1.2.2 The Pythagorean theorem


One of the most fundamen-
tal results is the well-known
Pythagorean Theorem. This
states thata^2 +b^2 =c^2 in a right
triangle with sides a and b and
hypotenuse c. The figure to the
right indicates one of the many
known proofs of this fundamental
result. Indeed, the area of the
“big” square is (a+b)^2 and can be
decomposed into the area of the
smaller square plus the areas of the
four congruent triangles. That is,


(a+b)^2 =c^2 + 2ab,

which immediately reduces toa^2 +b^2 =c^2.


Next, we recall the equally well-
known result that the sum of the
interior angles of a triangle is 180◦.
The proof is easily inferred from the
diagram to the right.


Exercises



  1. Prove Euclid’s Theorem for
    Proportional Segments, i.e.,
    given the right triangle 4 ABC as
    indicated, then


h^2 =pq, a^2 =pc, b^2 =qc.


  1. Prove that the sum of the interior angles of a quadrilateralABCD
    is 360◦.

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