How to find the sides of a right triangle? Fundamentals of geometry

The legs and the hypotenuse are the sides of a right triangle. The first are the segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is opposite the angle of 90 ^° . A Pythagorean triangle is the one whose sides are equal to natural numbers; Their length in this case is called the "Pythagorean troika".

The Egyptian Triangle

In order for the current generation to recognize geometry in the form in which it is taught in school now, it has evolved over several centuries. The fundamental point is the theorem of Pythagoras. The sides of a rectangular triangle (the figure is known to the whole world) are 3, 4, 5.

Few people are not familiar with the phrase "Pythagorean pants in all directions are equal." However, in fact, the theorem sounds like this: c ² (squared hypotenuse) = a ² + b ² (the sum of the squares of the legs).

Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called "Egyptian". Interestingly, the radius of the circle, which is inscribed in the figure, is equal to one. The name arose around the 5th century BC, when the philosophers of Greece traveled to Egypt.

When constructing the pyramids, architects and surveyors used the ratio 3: 4: 5. Such structures turned out to be proportional, pleasant in appearance and spacious, and also rarely collapsed.

In order to build a right angle, the builders used a rope, on which twelve knots were tied. In this case, the probability of constructing a rectangular triangle was raised to 95%.

Equality signs

• An acute angle in a right-angled triangle and a large side that 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 sharp angles are also equal. Thus, the triangles are the same in the second sign.
• When two figures are superimposed on each other, we rotate them so that they, having combined, become one isosceles triangle. According to its properties, the sides, or more precisely, the hypotenuse, are equal, as are the corners at the base, which means that these figures are the same.

By the first sign it is very simple to prove that the triangles are really equal, the main thing is that the two smaller sides (ie, the legs) are equal.

Triangles will be identical in the second sign, the essence of which is the equality of the leg and the acute angle.

Properties of a right angle triangle

The height that was lowered from the right angle, divides the figure into two equal parts.

The sides of a right-angled triangle and its medians are easy to recognize by the rule: the median, which is lowered to the hypotenuse, is equal to its half. The area of the 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-angled triangle, the properties of the angles are 30 ^° , 45 ^°, and 60 ^° .

• At an angle of 30 ^° , it should be remembered that the opposing leg will be 1/2 of the largest side.
• If the angle is 45 ^о , then the second acute angle is also 45 ^о . This suggests that the triangle is isosceles, and its legs are the same.
• The property of an angle of 60 ° is that the third angle has a degree measure of 30 °.

The area is easily recognized by one of three formulas:

1. Through the height and side, on which it sinks;
2. By Heron's formula;
3. On the sides and the corner between them.

The sides of a right-angled triangle, or more precisely the legs, converge with two heights. In order to find the third, it is necessary to consider the triangle formed, and then, by the Pythagorean theorem, calculate the required length. In addition to this formula, there is also a ratio of the doubled area and length of the hypotenuse. The most common expression among students is the first, since it requires less calculations.

Theorems applied to a right triangle

The geometry of a right-angled triangle includes the use of theorems such as:

1. 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 ratio is the key. You can use the formula if you have a triangle, for example, SNH. SN - hypotenuse, and it must be found. Then SN ² = NH ² + HS ² .
2. The cosine theorem. Generalizes the theorem of Pythagoras: g ² = f ² + s ² -2fs * cos the angle between them. For example, a DOB triangle is given. Known DB cathete and hypotenuse DO, it is necessary to find OB. Then the formula takes the given form: OB ² = DB ² + DO ² -2DB * DO * cos of the angle D. There are three consequences: the angle of the triangle will be acute, if the square of the third is subtracted from the sum of the squares of the two sides, the result should be less than zero. The angle is obtuse, if the expression is greater than zero. The angle is a straight line for zero.
3. The sine theorem. It shows the dependence of the sides on the opposite corners. In other words, this is the ratio of the lengths of the sides to the sinuses of the opposing corners. In the triangle HFB, where the hypotenuse is HF, there will be: HF / sin angle B = FB / sin angle H = HB / sin angle F.