The legs and the hypotenuse are the sides of a right triangle. The first are segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is opposite the angle at 90o. A Pythagorean triangle is one whose sides are equal to natural numbers; their lengths in this case are called the "Pythagorean triple".
Egyptian triangle
In order for the current generation to learn geometry in the form in which it is taught at school now, it has been developing for several centuries. The fundamental point is the Pythagorean theorem. The sides of a right triangle (the figure is known all over the world) are 3, 4, 5.
Few people are not familiar with the phrase "Pythagorean pants are equal in all directions." However, the theorem actually sounds like this: c2 (the square of the hypotenuse)=a2+b2 (the sum of the squares legs).
Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called "Egyptian". It is interesting that the radius of the circle, which is inscribed in the figure, is equal to one. The name originated around the 5th century BC, when Greek philosophers traveled to Egypt.
When constructing the pyramids, architects and surveyors used a ratio of 3:4:5. Such structures turned out to be proportional, pleasing to the eye and spacious, and also rarely collapsed.
In order to build a right angle, the builders used a rope on which 12 knots were tied. In this case, the probability of constructing a right-angled triangle increased to 95%.
Signs of equal figures
- An acute angle in a right triangle and a large side, which are equal to the same elements in the second triangle, is an indisputable sign of equality of figures. Taking into account the sum of the angles, it is easy to prove that the second acute angles are also equal. Thus, the triangles are identical in the second feature.
- When two figures are superimposed on each other, rotate them in such a way that they, combined, become one isosceles triangle. According to its property, the sides, or rather, the hypotenuses, are equal, as are the angles at the base, which means that these figures are the same.
By the first sign it is very easy to prove that the triangles are really equal, the main thing is that the two smaller sides (i.e. legs) are equal to each other.
Triangles will be the same in II feature, the essence of which is the equality of the leg and the acute angle.
Properties of a triangle with a right angle
The height lowered from the right angle splits the figure into two equal parts.
The sides of a right-angled triangle and its median are easy to recognize by the rule: the median, which is lowered to the hypotenuse, is equal to half of it. The area of a figure can be found both by Heron's formula and by the statement that it is equal to half the product of the legs.
In a right triangle, the properties of angles at 30o, 45o and 60o.
- With an angle that is 30o, remember that the opposite leg will be equal to 1/2 of the largest side.
- If the angle is 45o, then the second acute angle is also 45o. This suggests that the triangle is isosceles, and its legs are the same.
- The property of an angle of 60o is that the third angle has a degree measure of 30o.
The area is easy to find out by one of three formulas:
- through the height and side on which it falls;
- according to Heron's formula;
- on the sides and the angle between them.
The sides of a right-angled triangle, or rather the legs, converge with two heights. In order to find the third, it is necessary to consider the resulting triangle, and then, using the Pythagorean theorem, calculate the required length. In addition to this formula, there is also the ratio of twice the area and the length of the hypotenuse. The most common expression among students is the first, as it requires less calculations.
Theorems applied to a rectangulartriangle
The geometry of a right triangle includes the use of theorems such as:
- The Pythagorean theorem. Its essence lies in the fact that the square of the hypotenuse is equal to the sum of the squares of the legs. In Euclidean geometry, this relation is key. You can use the formula if a triangle is given, for example, SNH. SN is the hypotenuse and needs to be found. Then SN2=NH2+HS2.
- Cosine theorem. Generalizes the Pythagorean theorem: g2=f2+s2-2fscos of the angle between them. For example, given a triangle DOB. The leg DB and the hypotenuse DO are known, it is necessary to find OB. Then the formula takes this form: OB2=DB2+DO2-2DBDOcos angle D. There are three consequences: the angle of the triangle will be acute, if the square of the length of the third is subtracted from the sum of the squares of the two sides, the result must be less than zero. The angle is obtuse if this expression is greater than zero. Angle is a right angle when equal to zero.
- Sine theorem. It shows the relationship of sides to opposite angles. In other words, this is the ratio of the lengths of the sides to the sines of the opposite angles. In triangle HFB, where the hypotenuse is HF, it will be true: HF/sin of angle B=FB/sin of angle H=HB/sin of angle F.
