hypotenuse

A right triangle and its hypotenuse

In geometry , a hypotenuse is the longest side of a right triangle , which is always the side opposite the right angle . The length of the hypotenuse of a right triangle can be found using Pythagorean's Theorem , which says that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. For example, if one of the cathets is 3 cm (9 cm² square) and the other is 4 cm (16 cm² square), their squares add up to 25 cm². The length of the hypotenuse is the square root of 25 cm², which is 5 cm.

etymology

The word hypotenuse comes from the Greek .eta τὴν ὀρθὴν γωνίαν ὑποτείνουσα hē Ten ORTHEN gōnían hypoteínousa (sc. Γραμμή programs or πλευρά pleura ), it means "side opposite to the right angle" ( APOLLODORUS ). The nominalized participle, ἡ ὑποτείνουσα hē hypoteínousa , was used until the fourth century BC. Used for the hypotenuse of the triangle (documented in Plato , Timaios 54d). The Greek term was borrowed into late Latin in the form hypotēnūsa . The spelling with -e as a hypotenuse is of French origin ( Estienne de La Roche 1520 ).

Calculation of the hypotenuse

A right triangle

The length of the hypotenuse can be calculated using two specified lengths or a length and an acute angle .

Two catheters

For a right-angled triangle, if you name the length of the hypotenuse and the lengths of the cathetus and , according to the Pythagorean theorem : ${\ displaystyle c}$ ${\ displaystyle a}$ ${\ displaystyle b}$

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

If you solve this, you get the formula (under the condition ) ${\ displaystyle c}$ ${\ displaystyle c \ geq 0}$

{\ displaystyle c = {\ sqrt {a ^ {2} + b ^ {2}}},}

with which one can calculate the length of the hypotenuse.

Cathete and height

The height divides a right triangle into two sub-triangles. The base of the height divides the hypotenuse into the hypotenuse sections and . According to the Pythagorean theorem , so . The right triangle is similar to its part triangles because the three interior angles are the same. Therefore, the corresponding aspect ratios match and it applies , so ${\ displaystyle h}$ ${\ displaystyle c}$ ${\ displaystyle p}$ ${\ displaystyle q}$ ${\ displaystyle h ^ {2} + p ^ {2} = a ^ {2}}$ ${\ displaystyle p = {\ sqrt {a ^ {2} -h ^ {2}}}}$ ${\ displaystyle {\ frac {c} {a}} = {\ frac {a} {p}}}$

{\ displaystyle c = {\ frac {a ^ {2}} {p}} = {\ frac {a ^ {2}} {\ sqrt {a ^ {2} -h ^ {2}}}}}

and as well

{\ displaystyle c = {\ frac {b ^ {2}} {q}} = {\ frac {b ^ {2}} {\ sqrt {b ^ {2} -h ^ {2}}}}.}

Cathetus and acute angle

According to the definition of sine and cosine :

{\ displaystyle \ sin (\ alpha) = {\ frac {a} {c}}, \ quad \ cos (\ alpha) = {\ frac {b} {c}}, \ quad \ sin (\ beta) = {\ frac {b} {c}}, \ quad \ cos (\ beta) = {\ frac {a} {c}}}

{\ displaystyle c = {\ frac {a} {\ sin (\ alpha)}}, \ quad c = {\ frac {b} {\ cos (\ alpha)}}, \ quad c = {\ frac {b } {\ sin (\ beta)}}, \ quad c = {\ frac {a} {\ cos (\ beta)}}}

Height and acute angle

According to the definition of tangent and cotangent, the following applies to the sides and angles of the partial triangles:

{\ displaystyle \ tan (\ alpha) = {\ frac {p} {h}}, \ quad \ cot (\ alpha) = {\ frac {q} {h}}, \ quad \ tan (\ beta) = {\ frac {q} {h}}, \ quad \ cot (\ beta) = {\ frac {p} {h}}}

{\ displaystyle p = h \ cdot \ tan (\ alpha), \ quad q = h \ cdot \ cot (\ alpha), \ quad q = h \ cdot \ tan (\ beta), \ quad p = h \ cdot \ cot (\ beta)}

This results in the length of the hypotenuse

{\ displaystyle c = p + q = h \ cdot \ tan (\ alpha) + h \ cdot \ cot (\ alpha) = h \ cdot (\ tan (\ alpha) + \ cot (\ alpha))}

{\ displaystyle c = p + q = h \ cdot \ cot (\ beta) + h \ cdot \ tan (\ beta) = h \ cdot (\ cot (\ beta) + \ tan (\ beta))}

Many computer languages support the ISO-C standard function hypot(x, y), which returns the above value. The function is designed in such a way that it does not fail even if the simple calculation according to the formula can overflow or underflow, and is also often somewhat more precise.

Some scientific calculators offer a function to convert Cartesian coordinates to polar coordinates . This outputs both the length of the hypotenuse and the angle that the hypotenuse forms with the baseline if and are given. The angle returned is usually given by. ${\ displaystyle x}$ ${\ displaystyle y}$ arctan2(y, x)

properties

Orthogonal projections :

The length of the hypotenuse is the sum of the lengths of the orthogonal projections of both cathets .

{\ displaystyle c = p + q}

The height divides a right triangle into two sub-triangles. The right triangle is similar to its sub-triangles because the three interior angles are the same (see illustration). Therefore the following aspect ratios apply:

{\ displaystyle {\ frac {a} {c}} = {\ frac {p} {a}}}

{\ displaystyle {\ frac {b} {c}} = {\ frac {q} {b}}}

The square of the length of a leg is the product of the length of its orthogonal projection and the length of the hypotenuse.

{\ displaystyle a ^ {2} = p \ cdot c}

{\ displaystyle b ^ {2} = q \ cdot c}

So the length of a leg is the geometric mean between the length of its orthogonal projection and the length of the hypotenuse.

{\ displaystyle a = {\ sqrt {p \ cdot c}}}

{\ displaystyle b = {\ sqrt {q \ cdot c}}}

Trigonometric functions

With the help of trigonometric functions one can calculate the values of the two acute angles and the right triangle . ${\ displaystyle \ alpha}$ ${\ displaystyle \ beta}$

The lengths of the hypotenuse and the cathetus are given , the ratio of which applies: ${\ displaystyle c}$ ${\ displaystyle b}$

{\ displaystyle {\ frac {b} {c}} = \ sin (\ beta)}

The inverse trigonometric function is

{\ displaystyle \ beta = \ arcsin \ left ({\ frac {b} {c}} \ right),}

in which the angle is opposite the cathetus . The adjacent angle of the cathetus is one can also use the formula to change the size of the angle ${\ displaystyle \ beta}$ ${\ displaystyle b}$ ${\ displaystyle b}$ ${\ displaystyle \ alpha = 90 ^ {\ circ} - \ beta.}$ ${\ displaystyle \ beta}$