People became interested in surprising properties of rectangular triangles at the time of antiquity. Many of these properties were described by the Ancient Greek scientist Pythagoras. In Ancient Greece also names of the parties of a rectangular triangle appeared.
What triangle is called rectangular?
There are several types of triangles. At some all acute angles, at others – one stupid and two sharp, at the third – two sharp and a straight line. On this sign each type of these geometrical figures also received the name: acute-angled, obtusangular and rectangular. That is, such triangle at which one of corners makes 90 ° is called rectangular. There is also other definition similar with the first. The triangle at which two parties are perpendicular is called rectangular.
Hypotenuse and legs
At acute-angled and obtusangular triangles the pieces connecting tops of corners are called just the parties. At a triangle rectangular the parties have also other names. Those which adjoin to a right angle are called legs. The party, opposite to a right angle, is called a hypotenuse. In translation from Greek the word "hypotenuse" means "tense", and "leg" - "perpendicular".
Ratios between a hypotenuse and legs
The parties of a rectangular triangle are connected among themselves by certain ratios which considerably facilitate calculations. For example, knowing the sizes of legs, it is possible to calculate hypotenuse length. This ratio by name the mathematician who opened it received the name of Pythagorean theorem and it looks so:
c2=a2+b2, where with – a hypotenuse, an and b – legs. That is, the hypotenuse will be equal to a square root from the sum of squares of legs. To find any of legs, it is enough to subtract a square of other leg from a square of a hypotenuse and to take a square root from the received difference.
Adjacent and opposite leg
Draw a rectangular triangle of DIA. By letter C it is accepted to designate top of a right angle, And yes In – tops of acute angles. The parties, opposite to each corner, it is convenient to call and, b and with, by names of the corners lying opposite to them. Consider A. Katet's corner and for it there will be opposite, b leg – adjacent. The relation of an opposite leg to a hypotenuse is called a sine. It is possible to calculate this trigonometrical function on a formula: sinA=a/c. The relation of an adjacent leg to a hypotenuse is called a cosine. He is found on a formula: cosA=b/c. Thus, knowing a corner and one of the parties, it is possible to calculate other party on these formulas. Trigonometrical ratios connected also both legs. The relation opposite to adjacent is called a tangent, and adjacent to opposite – a cotangent. It is possible to express these ratios formulas tgA=a/b or ctgA=b/a.