A triangle is a geometric number made up of three segments that connect three points that do not lie on the same line. The points that form a triangle are called its points, and the segments are side by side.
Depending on the type of triangle (rectangular, monochrome, etc.) you can calculate the side of the triangle in different ways, depending on the input data and the conditions of the problem.
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To calculate the sides of a right triangle, the Pythagorean theorem is used, according to which the square of the hypotenuse is equal to the sum of the squares of the leg.
If we label the legs with "a" and "b" and the hypotenuse with "c", then pages can be found with the following formulas:
If the acute angles of a right triangle (a and b) are known, its sides can be found with the following formulas:
cropped triangle
A triangle is called an equilateral triangle in which both sides are the same.
How to find the hypotenuse in two legs
If the letter "a" is identical to the same page, "b" is the base, "b" is the corner opposite the base, "a" is the adjacent corner, the following formulas can be used to calculate pages:
Two corners and side
If one page (c) and two angles (a and b) of any triangle are known, the sine formula is used to calculate the remaining pages:
You must find the third value y = 180 - (a + b) because
the sum of all the angles of a triangle is 180°; 
Two sides and an angle
If two sides of a triangle (a and b) and the angle between them (y) are known, the cosine theorem can be used to calculate the third side.
How to determine the perimeter of a right triangle
A triangular triangle is a triangle, one of which is 90 degrees, and the other two are acute. calculation perimeter such triangle depending on the amount of known information about it.
You will need it
- Depending on the occasion, skills 2 of the three sides of the triangle, as well as one of its sharp corners.
instructions
the first Method 1. If all three pages are known triangle. Then, whether perpendicular or not triangular, the perimeter is calculated as: P = A + B + C, where possible, c is the hypotenuse; a and b are legs.
second Method 2.
If a rectangle has only two sides, then using the Pythagorean theorem, triangle can be calculated using the formula: P = v (a2 + b2) + a + b or P = v (c2 - b2) + b + c.
third Method 3. Let the hypotenuse be c and an acute angle? Given a right triangle, it will be possible to find the perimeter in this way: P = (1 + sin?
fourth Method 4. They say that in the right triangle the length of one leg is equal to a and, on the contrary, has an acute angle. Then calculate perimeter this is triangle will be performed according to the formula: P = a * (1 / tg?
1 / son? + 1)
fifth Method 5.
Triangle Online Calculation
Let our leg lead and be included in it, then the range will be calculated as: P = A * (1 / CTG + 1 / + 1 cos?)
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The Pythagorean theorem is the basis of any mathematics. Specifies the relationship between the sides of a true triangle. Now there are 367 proofs of this theorem.
instructions
the first The classic school formulation of the Pythagorean theorem sounds like this: the square of the hypotenuse is equal to the sum of the squares of the legs.
To find the hypotenuse in a right triangle of two Catets, you must turn to square the length of the legs, assemble them, and take the square root of the sum. In the original formulation of his statement, the market is based on the hypotenuse, equal to the sum of the squares of 2 squares produced by Catete. However, the modern algebraic formulation does not require the introduction of a domain representation.
second For example, a right triangle whose legs are 7 cm and 8 cm.
Then, according to the Pythagorean theorem, the square hypotenuse is R + S = 49 + 64 = 113 cm. The hypotenuse is equal to the square root of 113.
Angles of a right triangle
The result was an unreasonable number.
third If the triangles are legs 3 and 4, then the hypotenuse = 25 = 5. When you take the square root, you get a natural number. The numbers 3, 4, 5 form a Pygagorean triple, since they satisfy the relation x? +Y? = Z, which is natural.
Other examples of a Pythagorean triplet are: 6, 8, 10; 5, 12, 13; 15, 20, 25; 9, 40, 41.
fourth In this case, if the legs are identical to each other, the Pythagorean theorem turns into a more primitive equation. For example, let such a hand be equal to the number A and the hypotenuse is defined for C, and then c? = Ap + Ap, C = 2A2, C = A? 2. In this case, you don't need A.
fifth The Pythagorean theorem is a special case that is larger than the general cosine theorem, which establishes a relationship between the three sides of a triangle for any angle between two of them.
Tip 2: How to determine the hypotenuse for legs and angles
The hypotenuse is called the side in a right triangle that is opposite the 90 degree angle.
instructions
the first In the case of well-known catheters, as well as an acute angle of a right triangle, the hypotenuse can have a size equal to the ratio of the leg to the cosine / sine of this angle, if the angle was opposite / e include: H \u003d C1 (or C2) / sin, H \u003d C1 (or С2 ?) / cos ?. Example: Let ABC be given an irregular triangle with hypotenuse AB and right angle C.
Let B be 60 degrees and A 30 degrees. The length of the stem BC is 8 cm. The length of the hypotenuse AB should be found. To do this, you can use one of the above methods: AB = BC / cos60 = 8 cm. AB = BC / sin30 = 8 cm.
The hypotenuse is the longest side of the rectangle triangle. It is located at a right angle. Method for finding the hypotenuse of a rectangle triangle depending on the source data.
instructions
the first If your legs are perpendicular triangle, then the length of the hypotenuse of the rectangle triangle can be found by the Pythagorean analogue - the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs: c2 = a2 + b2, where a and b are the length of the legs of the right triangle .
second If it is known and one of the legs is at an acute angle, the formula for finding the hypotenuse will depend on the presence or absence at a certain angle with respect to the known leg - adjacent (the leg is located near), or vice versa (the opposite case is located nego.V of the specified angle is equal to the fraction leg hypotenuse in cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio of sinusoidal angles: da = a / sin.
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Helpful Hints
An angular triangle whose sides are connected as 3:4:5, called the Egyptian delta, due to the fact that these figures were widely used by the architects of ancient Egypt.
This is also the simplest example of Jeron's triangles, with pages and area represented as integers.
A triangle is called a rectangle whose angle is 90°. The side opposite the right corner is called the hypotenuse, the other side is called the legs.
If you want to find how a right triangle is formed by some properties of regular triangles, namely the fact that the sum of the acute angles is 90°, which is used, and the fact that the length of the opposite leg is half the hypotenuse is 30°.
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cropped triangle
One of the properties of an equal triangle is that its two angles are the same.
To calculate the angle of a right equilateral triangle, you need to know that:
- It's no worse than 90°.
- The values of acute angles are determined by the formula: (180 ° -90 °) / 2 = 45 °, i.e.
Angles α and β are 45°.
If the known value of one of the acute angles is known, the other can be found using the formula: β = 180º-90º-α or α = 180º-90º-β.
This ratio is most commonly used if one of the angles is 60° or 30°.
Key Concepts
The sum of the interior angles of a triangle is 180°.
Because it's one level, two stay sharp.
Calculate triangle online
If you want to find them, you need to know that:
other methods
The acute angle values of a right triangle can be calculated from the mean - with a line from a point on the opposite side of the triangle, and the height - the line is a perpendicular drawn from the hypotenuse at a right angle.
Let the median extend from the right corner to the middle of the hypotenuse, and h be the height. In this case it turns out that: 
- sinα = b / (2 * s); sin β = a / (2 * s).
- cosα = a / (2 * s); cos β = b / (2 * s).
- sinα = h / b; sin β = h / a.
Two pages
If the lengths of the hypotenuse and one of the legs are known in a right triangle or from two sides, then trigonometric identities are used to determine the values of acute angles:
- α=arcsin(a/c), β=arcsin(b/c).
- α=arcos(b/c), β=arcos(a/c).
- α = arctan (a / b), β = arctan (b / a).
Length of a right triangle
Area and Area of a Triangle
perimeter
The circumference of any triangle is equal to the sum of the lengths of the three sides. The general formula for finding a triangular triangle is:
where P is the circumference of the triangle, a, b and c are its sides.
Perimeter of an equal triangle can be found by successively combining the lengths of its sides, or multiplying the side length by 2 and adding the length of the base to the product.
The general formula for finding an equilibrium triangle will look like this:
where P is the perimeter of an equal triangle, but either b, b are the base.
Perimeter of an equilateral triangle can be found by successively combining the lengths of its sides, or by multiplying the length of any page by 3.
The general formula for finding the rim of equilateral triangles would look like this:
where P is the perimeter of an equilateral triangle, a is any of its sides.
region
If you want to measure the area of a triangle, you can compare it to a parallelogram. Consider triangle ABC:
If we take the same triangle and fix it so that we get a parallelogram, we get a parallelogram with the same height and base as this triangle:
In this case, the common side of the triangles is folded together along the diagonal of the molded parallelogram.
From the properties of a parallelogram. It is known that the diagonals of a parallelogram are always divided into two equal triangles, then the surface of each triangle is equal to half the range of the parallelogram.
Since the area of the parallelogram is the product of its base height, the area of the triangle will be half that product. So for ΔABC the area will be the same
Now consider a right triangle:
Two identical right triangles can be bent into a rectangle if it leans against them, which is every other hypotenuse.
Since the surface of the rectangle coincides with the surface of the adjacent sides, the area of this triangle is the same:
From this we can conclude that the surface of any right triangle is equal to the product of the legs divided by 2.
From these examples, we can conclude that the surface of each triangle is the same as the product of the length, and the height is reduced to the base divided by 2.
The general formula for finding the area of a triangle would look like this:
where S is the area of the triangle, but its base, but the height falls to the bottom a.
A triangle is called a right triangle if one of its angles is 90º. The side opposite the right angle is called the hypotenuse, and the other two are the legs.
To find the angle in a right triangle, some properties of right triangles are used, namely: the fact that the sum of acute angles is 90º, and also the fact that opposite the leg, the length of which is half the hypotenuse, lies an angle equal to 30º.
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Isosceles triangle
One of the properties of an isosceles triangle is that two of its angles are equal. To calculate the values of the angles of a right-angled isosceles triangle, you need to know that:
- A right angle is 90º.
- The values of acute angles are determined by the formula: (180º-90º)/2=45º, i.е. angles α and β are 45º.
If the value of one of the acute angles is known, the second one can be found by the formula: β=180º-90º-α, or α=180º-90º-β. Most often, this ratio is used if one of the angles is 60º or 30º.
Key Concepts
The sum of the interior angles of a triangle is 180º. Since one angle is right, the other two will be sharp. To find them, you need to know that:
other methods
The values of the acute angles of a right triangle can be calculated by knowing the value of the median - a line drawn from the vertex to the opposite side of the triangle, and the height - a straight line, which is a perpendicular dropped from the right angle to the hypotenuse. Let s be the median drawn from the right angle to the midpoint of the hypotenuse, h be the height. In this case it turns out that: 
- sinα=b/(2*s); sinβ=a/(2*s).
- cosα=a/(2*s); cos β=b/(2*s).
- sinα=h/b; sinβ=h/a.
Two sides
If the lengths of the hypotenuse and one of the legs, or two sides, are known in a right triangle, trigonometric identities are used to find the values of acute angles:
- α=arcsin(a/c), β=arcsin(b/c).
- α=arcos(b/c), β=arcos(a/c).
- α=arctg(a/b), β=arctg(b/a).
In geometry, there are often problems related to the sides of triangles. For example, it is often necessary to find the side of a triangle if the other two are known.
Triangles are isosceles, equilateral and equilateral. From all the variety, for the first example, we choose a rectangular one (in such a triangle, one of the angles is 90 °, the sides adjacent to it are called the legs, and the third is the hypotenuse).
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The length of the sides of a right triangle
The solution of the problem follows from the theorem of the great mathematician Pythagoras. It says that the sum of the squares of the legs of a right triangle is equal to the square of its hypotenuse: a²+b²=c²
- Find the square of the leg length a;
- Find the square of the leg b;
- We put them together;
- From the result obtained, we extract the root of the second degree.
Example: a=4, b=3, c=?
- a²=4²=16;
- b²=3²=9;
- 16+9=25;
- √25=5. That is, the length of the hypotenuse of this triangle is 5.
If the triangle does not have a right angle, then the lengths of the two sides are not enough. This requires a third parameter: it can be an angle, height, area of a triangle, radius of a circle inscribed in it, etc.
If the perimeter is known
In this case, the task is even easier. The perimeter (P) is the sum of all sides of the triangle: P=a+b+c. Thus, by solving a simple mathematical equation, we get the result.
Example: P=18, a=7, b=6, c=?
1) We solve the equation, transferring all known parameters to one side of the equal sign:
2) Substitute values instead of them and calculate the third side:
c=18-7-6=5, total: the third side of the triangle is 5.
If the angle is known
To calculate the third side of a triangle given the angle and the other two sides, the solution is reduced to calculating the trigonometric equation. Knowing the relationship of the sides of the triangle and the sine of the angle, it is easy to calculate the third side. To do this, you need to square both sides and add their results together. Then subtract from the resulting product of the sides, multiplied by the cosine of the angle: C=√(a²+b²-a*b*cosα)
If the area is known
In this case, one formula is not enough.
1) First, we calculate sin γ by expressing it from the formula for the area of a triangle:
sin γ= 2S/(a*b)
2) Using the following formula, we calculate the cosine of the same angle:
sin² α + cos² α=1
cos α=√(1 - sin² α)=√(1- (2S/(a*b))²)
3) And again we use the sine theorem:
C=√((a²+b²)-a*b*cosα)
C=√((a²+b²)-a*b*√(1- (S/(a*b))²))
Substituting the values of the variables into this equation, we obtain the answer to the problem.
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 90 degree angle. 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 developed for several centuries. The fundamental point is the Pythagorean theorem. The sides of a rectangle are known to the whole world) are 3, 4, 5.
Few people are not familiar with the phrase "Pythagorean pants are equal in all directions." However, in fact, the theorem sounds like this: c 2 (the square of the hypotenuse) \u003d 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". It is interesting that which is inscribed in the figure is equal to one. The name arose around the 5th century BC, when Greek philosophers traveled to Egypt.
When building the pyramids, architects and surveyors used the ratio 3:4:5. Such structures turned out to be proportional, pleasant to look at 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 equality of 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 the equality of the 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 criterion.
- When two figures are superimposed on each other, we rotate them in such a way that, when combined, they become one isosceles triangle. According to its property, the sides, or rather, the hypotenuses, are equal, as well as 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., the legs) are equal to each other.
The triangles will be the same according to the II sign, 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 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. 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 of 30 o, 45 o and 60 o apply.
- At an angle that is 30 °, it should be remembered 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 60 degrees is that the third angle has a measure of 30 degrees.
The area is easy to find by one of three formulas:
- through the height and the side on which it descends;
- according to Heron's formula;
- along the sides and the angle between them.
The sides of a right 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 that apply to a right triangle
The geometry of a right triangle includes the use of theorems such as:
