The legs and hypotenuse are sides of a right triangle. The first are segments that are adjacent to a right angle, and the hypotenuse is the longest part of the figure and is located opposite an angle of 90 about . 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 recognize 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-angled triangle (a figure known throughout 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: c 2 (squared hypotenuse) = a 2 + b 2 (sum of squared legs).
Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called "Egyptian." Interestingly, the radius of the circle that is inscribed in the figure is equal to unity. The name appeared around the 5th century BC, when the philosophers of Greece traveled to Egypt.
When building the pyramids, architects and surveyors used a 3: 4: 5 ratio. Such structures turned out to be proportional, pleasant-looking and spacious, and also rarely collapsed.
In order to build a right angle, builders used a rope on which 12 knots were tied. In this case, the probability of constructing a right triangle increased to 95%.
Signs of equality of figures
- An acute angle in a right-angled triangle and a large side that are equal to the same elements in the second triangle are an indisputable sign of equality of figures. Given the sum of the angles, it is easy to prove that the second acute angles are also equal. Thus, the triangles are the same in the second feature.
- When two figures are superimposed on each other, we turn them so that they, together, become one isosceles triangle. By its property, the sides, and more precisely, 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 simple to prove that the triangles are really equal, the main thing is that the two smaller sides (i.e. the legs) are equal to each other.
Triangles will be the same according to the II criterion, the essence of which is the equality of the leg and the acute angle.
Right Angle Triangle Properties
The height, which was lowered from a right angle, divides the figure into two equal parts.
The sides of a right-angled triangle and its median can be easily recognized by the rule: the median, which is omitted by 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 angles of 30 ° , 45 ° and 60 ° apply.
- At an angle of 30 ° , it should be remembered that the opposite leg will be 1/2 of the largest side.
- If the angle is 45 ° , then the second acute angle is also 45 ° . This suggests that the triangle is isosceles, and its legs are the same.
- The property of the angle of 60 about is that the third angle has a degree measure of 30 about .
The area is easy to recognize by one of three formulas:
- through the height and the side on which it falls;
- according to the Heron formula;
- on the sides and the corner between them.
The sides of a right-angled triangle, or rather the legs, converge with two heights. In order to find the third one, it is necessary to consider the formed triangle, and then by the Pythagorean theorem calculate the necessary length. In addition to this formula, there is also a ratio of doubled area to hypotenuse length. The most common expression among students is the first, as it requires fewer calculations.
Theorems Applied to a Right Triangle
The geometry of a right triangle includes the use of theorems such as:
- 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 key. You can use the formula if a triangle is given, for example, SNH. SN is the hypotenuse, and it must be found. Then SN 2 = NH 2 + HS 2 .
- Cosine theorem. Generalizes the Pythagorean theorem: g 2 = f 2 + s 2 -2fs * cos of the angle between them. For example, given the triangle DOB. Known DB leg and hypotenuse DO, you must find OB. Then the formula takes this 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-angled, 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 this expression is greater than zero. The angle is a straight line if it is equal to zero.
- Sine theorem. It shows the dependence of the parties to opposite corners. In other words, this is the ratio of the lengths of the sides to the sines of opposite angles. In the triangle HFB, where the hypotenuse is HF, it will be true: HF / sin of the angle B = FB / sin of the angle H = HB / sin of the angle F.