Math Pythagorean theorem Altitude-on-hypotenuse theorem

Altitude-on-hypotenuse theorem

If you put the altitude on the hypotenuse, it divides a triangle into two individual triangles, which are each right-angled. With the help of the Pythagorean theorem, further properties are derived. That is why it is called Altitude-on-hypotenuse theorem.

A square of cathetus of a right triangle is the same size as the rectangle of hypotenuse and the adjacent hypotenuse section.



  1. Find the right angle
  2. Drag the height through the right angle and divide the hypotenuse into 2 sections
  3. Change the formula appropriately, so that the side, you are looking for, is alone
  4. If a cathetus is being searched: Pull the square root ($\sqrt{}$) from the result


In the triangle ABC with $\gamma=90^\circ$ the altitude-on-hypotenuse theorem is:

$p\cdot c=a^2$

  • $a^2=p\cdot c$ $\Leftrightarrow$ $a=\sqrt{p\cdot c}$

  • $p=\frac{a^2}{c}$

  • $c=\frac{a^2}{p}$
such as

$q\cdot c=b^2$

  • $b^2=q\cdot c$ $\Leftrightarrow$ $b=\sqrt{q\cdot c}$

  • $q=\frac{b^2}{c}$

  • $c=\frac{b^2}{q}$