U16 Pythagoras' Theorem Essential Formulas

Pythagoras' Theorem

In any right-angled triangle, the square of the length of the hypotenuse $c$ is equal to the sum of the squares of the lengths of the other two sides $a$ and $b$.

Where $c$ is the length of the side opposite the right angle (the hypotenuse), and $a$ and $b$ are the lengths of the other two sides (the legs).

Finding Different Sides

The formula can be rearranged to find the length of any side when the other two are known.

To find the hypotenuse $c$.

To find a leg $a$ (similarly, $b = \sqrt{c^2 - a^2}$).

Testing for a Right Angle

If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is right-angled. The right angle is opposite the longest side.

For a triangle with sides of length $p$, $q$, and $r$, where $r$ is the longest, if $p^2 + q^2 = r^2$, then the triangle is right-angled with the right angle opposite side $r$.

Memorise These Triples

Sets of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. Recognising them can save calculation time.

• $(3, 4, 5)$ and its multiples e.g., $(6, 8, 10)$, $(9, 12, 15)$.
• $(5, 12, 13)$
• $(7, 24, 25)$
• $(8, 15, 17)$