Capítulo 14 · Neste capítulo você vai

Right triangles

  • identify hypotenuse, cathets, projections and height relative to the hypotenuse;
  • deduce metric relations from the similarity of triangles;
  • demonstrate and apply the Pythagorean theorem and its reciprocal;
  • calculate the diagonal of the square and the height of equilateral triangle;
  • use sine, cosine and tangent of 30°, 45° and 60°;
  • generate Pythagorean triangles, with integer sides.

The right triangle is the most used piece of applied geometry: every time a distance is measured “in a straight line” from two perpendicular measurements — the diagonal of a screen, the length of a ramp, the distance between two points on a map — it is Pythagoras’ theorem that works. In this chapter, the similarity of triangles from chapter 13 explains where it comes from.

The squares built on the legs fill the square built on the hypotenuse.
The squares built on the legs fill the square built on the hypotenuse.
A visual proof of the Pythagorean theorem by rearranging parts.
A visual proof of the Pythagorean theorem by rearranging parts.

For the more intuitive side of the theorem, with squares and numerical examples, see also the class Pythagorean Theorem of Elementary II.