Right triangle formulas
Pythagorean theoremIn 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 anglesIn 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 areaThe 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 theoremh : 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 ruleThe 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)`