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