Right triangle

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Right triangle formulas

Pythagorean theorem

In a right triangle, the square of the hypotenus is equal to the sum of squares of the other two sides.

`c^2 = a^2+b^2`

The converse is also true. For a given triangle, if the square of the longest side is equal to the sum of squares of the other two sides then this triangle is right-angled.

Trigonometric ratios in a right triangle

`sin(\alpha) = b / c` ; `cos(\alpha) = a / c`

`tan(\alpha) = b / a` ; `cot(\alpha) = a / b`

Less used ratios are secant (sec) and cosecant (csc).

`sec(\alpha) = c / a = 1/cos(\alpha)` ; `csc(\alpha) = c / b = 1/sin(\alpha)`

In other words,

sine = opposite side / hypotenuse
cosine = adjacent side / hypotenuse
tangent = opposite side / adjacent side
cotangent = adjacent side / opposite side

And for less used ratios,
secant = hypotenuse / adjacent side
cosecant = hypotenuse / opposite side

Complementary angles

In a right triangle, the acute angles `\alpha` and `\beta` are complementary because the sum of the three angles is 180 degrees and the third (right) angle is 90 degrees. So,

`\alpha + \beta = 90°`

Perimeter and area

The perimeter of a triangle is simply equal to the sum of its three sides.
`P = a+b+c`

The area of the right triangle is equal to, `A = (a*b)/2`

Right triangle altitude theorem

h : altitude (or height) on the hypotenuse
p : projection of leg a on the hypotenuse
q : projection of leg b on the hypotenuse

The square of the altitude on the hypotenuse is equal to the geometric mean of the projections of the legs (non-hypotenuse sides) on the hypotenuse.

`h^2 = p*q`

Leg geometric mean Theorem or Leg rule

The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the projection of that leg on the hypotenuse.

`a^2 = p*c <=> a = sqrt(p*c)`
`b^2 = q*c <=> b = sqrt(q*c)`