EQN-04-09

Pythagoras & Special Right Triangles

Apply the Pythagorean theorem, its famous triples, and the two special-angle ratios to find any missing side.

~6 min read•Difficulty 3/5

For a right triangle with legs a, b and hypotenuse c (the side opposite the right angle):

a^{2} + b^{2} = c^{2}

The hypotenuse is always the longest side.

3 - 4 - 5, 5 - 12 - 13, 8 - 15 - 17, 7 - 24 - 25

Any multiple works too (6-8-10, 9-12-15). Spotting a triple avoids a square-root computation.

45 - 45 - 90 : legs equal, sides in ratio 1 : 1 : 2

30 - 60 - 90 : sides in ratio 1 : 3 : 2 (short : long : hypotenuse)

From any one side you can get the rest by scaling the ratio.

Diagonal of a square of side $s = s 2 (a 45$ –45–90 triangle).
Height of an equilateral triangle of side $s = (s 3) /2 (a 30$ –60–90 triangle).

Diagram

Pythagoras $: a^{2} + b^{2} = c^{2} ($ the 9-12-15 triple).

Worked examples

Example 1 (difficulty 2/5)

A right triangle has legs 9 and 12. What is the hypotenuse?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

What is the diagonal of a square with side 6?

◎ Predict the answer first:

Adding the legs

(9 + 12)

to get the hypotenuse

Why: Perimeter thinking creeps in.

Fix: Use

a^{2} + b^{2} = c^{2},

not

a + b .

Swapping the 30–60–90 ratio order

Why: Three different sides are easy to mis-assign.

Fix: Short : long : hyp

= 1 : 3 : 2

— short faces 30°, hyp faces 90°.