Leg

A leg, in geometry, is any of the two smaller sides of a right triangle, those that make up the right angle. Its name comes from the Latin cathetus, a loan from the Greek κάθετος, káthetos ('vertical, perpendicular').

The side with the longest measure is called the hypotenuse – the one opposite the right angle. The denomination of legs and hypotenuse applies exclusively to the sides of right triangles.

Properties of legs

Pythagorean Theorem

The square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs.

c2=a2+b2{displaystyle c^{2}=a^{2}+b^{2}}

In the figure, the sides a and b are the legs and c the hypotenuse. Let's see it with an example:

Imagine that side a is 5 cm and side b is 4 cm and you want to calculate the hypotenuse (side c). Then it would be done:

5² + 4² = 25 + 16 = 41

The value of the hypotenuse would be equal to the square root of 41.

Orthogonal projections

The square of the length of a leg is equal to the product of its orthogonal projection on the hypotenuse by its length.

a2=c⋅ ⋅ nb2=c⋅ ⋅ m{displaystyle {begin{aligned}a^{2}= stranger,ccdot nb^{2}= stranger,ccdot mend{aligned}}}}}}}}}

That is, the length of a leg a is mean proportional between the lengths of its projection n and that of the hypotenuse c.

ca=ancb=bm{displaystyle {begin{aligned}{frac}{c}{a}}}}{ fake{a}{n}}{frac {c}{b}}{b}}}}}{b}{b}{b}{m}}{aligned}}}}

In the figure, the hypotenuse is side c and the legs are sides a and b. The orthogonal projection of a' is n, and that of b is m.

Trigonometric ratios

By means of trigonometric ratios you can obtain the value of the acute angles of the right triangle. With respect to an angle, a leg is called adjacent or contiguous, if it forms the angle together with the hypotenuse, and opposite if it is not part of the given angle.

Known the length of the caetos b{displaystyle b,} and a{displaystyle a,}The reason between them is:

ba=So... (β β ){displaystyle {frac {b}{a}}=tan(beta),}

Therefore, the inverse trigonometric function is as follows:

β β =arctan (ba){displaystyle beta =arctan left({frac {b}{a}{a}right)},

being β β {displaystyle beta ,} the value of the angle opposite to the cateto b{displaystyle b,}.

The opposite angle of the cateto a{displaystyle a,}, called α α {displaystyle alpha ,}, will have the value:

α α =90 − − β β {displaystyle alpha =90^{circ }-beta ,}