Instructions

The angles opposite to the legs a and b are denoted by A and B, respectively. The hypotenuse, by definition, is the side of a right-angled triangle that is opposite to the right angle (while the hypotenuse forms acute angles with the other sides of the triangle). The length of the hypotenuse is denoted by s.

You will need:
Calculator.

Use the following expression for the leg: a = sqrt (c ^ 2-b ^ 2), if you know the values ​​of the hypotenuse and the other leg. This expression is obtained from the Pythagorean theorem, which states that the square of the hypotenuse of a triangle is equal to the sum of the squares of the legs. The sqrt statement stands for square root extraction. The "^ 2" sign means raising to the second power.

Use the formula a = c * sinA if you know the hypotenuse (c) and the angle opposite the desired leg (we denoted this angle as A).
Use the expression a = c * cosB to find the leg, if you know the hypotenuse (c) and the angle adjacent to the desired leg (we designated this angle as B).
Calculate the leg by the formula a = b * tgA in the case when leg b and the angle opposite to the desired leg are given (we agreed to designate this angle as A).

Note:
If in your task the leg is not found in any of the described ways, most likely, it can be reduced to one of them.

Helpful hints:
All these expressions are obtained from the well-known definitions of trigonometric functions, therefore, even if you forgot one of them, you can always quickly derive it by simple operations. Also, it is useful to know the values ​​of trigonometric functions for the most typical angles of 30, 45, 60, 90, 180 degrees.

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 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 differently, 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 mark the legs with the letters "a" and "b" and the hypotenuse with "c", then the pages can be found with the following formulas:

If the acute angles of a right-angled 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 to the base, "a" is the adjacent corner, the following formulas can be used to calculate the 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 should find the third value y = 180 - (a + b) because

the sum of all the angles of the triangle is 180 °;

Two sides and an angle

If you know the two sides of the triangle (a and b) and the angle between them (y), 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 sharp. calculation perimeter such triangle depending on the amount of known information about it.

You need it

• Depending on the case, skills are 2 of the three sides of the triangle, as well as one of its sharp corners.

instructions

first Method 1. If all three pages are known triangle Then, regardless, perpendicular or non-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 the rectangle has only two sides, then using the Pythagorean theorem, triangle can be calculated by the formula: P = v (a2 + b2) + a + b or P = v (c2 - b2) + b + c.

the third Method 3. Let the hypotenuse c and an acute angle? Given a right-angled triangle, it will be possible to detect the perimeter in this way: P = (1 + sin?

fourth Method 4. It is said 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 triangle will be performed according to the formula: P = a * (1 / tg?

1 / son? + 1)

fifth Method 5.

Online triangle 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 foundation of any mathematics. Defines the relationship between the sides of a true triangle. Now 367 proofs of this theorem are indicated.

instructions

first The classical 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, collect them, and take the square root of the sum. In the original formulation of his statement, the market is based on a hypotenuse equal to the sum 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-angled triangle whose legs are 7 cm and 8 cm.

Then, according to the Pythagorean theorem, the square hypotenuse is equal to R + S = 49 + 64 = 113 cm.The hypotenuse is equal to the square root of the number 113.

Angles of a right triangle

The result was an unreasonable number.

the 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 Pyghagorean triplet, since they satisfy the relation x? + Y? = Z, which is natural.

Other examples of the 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, suppose such a hand is 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 do not need A.

fifth The Pythagorean theorem is a special case that is larger than the general cosine theorem, which establishes a connection 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 to the 90 degree angle.

instructions

first In the case of known catheters, as well as an acute angle of a right-angled triangle, there may be a hypotenuse size equal to the ratio of the leg to the cosine / sine of this angle, if the angle was opposite / e include: H = C1 (or C2) / sin, H = C1 (or C2?) / Cos?. Example: Let ABC be an irregular triangle with hypotenuse AB and right angle C.

Let B be 60 degrees and A 30 degrees. BC stem length 8 cm. AB hypotenuse length should be found. To do this, you can use one of the above methods: AB = BC / cos60 = 8 cm. AB = BC / sin30 = 8 cm.

Hypotenuse is the longest side of the rectangle triangle... It is located at right angles. Rectangle hypotenuse search method triangle depending on the source data.

instructions

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 of nego is located. the hypotenuse of the leg at the cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio of the sinusoidal angles: da = a / sin.

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Useful Tips
An angular triangle, the sides of which 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 with an angle of 90 °. The side opposite to the right corner is called the hypotenuse, the other side is called the legs.

If you want to find how a right-angled triangle is formed by some of the 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 corners are the same.

To calculate the angle of a right-angled equal triangle, you need to know that:

• This is no worse than 90 °.
• Acute angle values ​​are determined by the formula: (180 ° -90 °) / 2 = 45 °, i.e.

  The angles α and β are equal to 45 °.

If the known value of one of the acute angles is known, the other can be found by the formula: β = 180º-90º-α or α = 180º-90º-β.

This ratio is most often used when one of the angles is 60 ° or 30 °.

Key concepts

The sum of the interior angles of a triangle is 180 °.

Because this is one level, two remain 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 dropped from the hypotenuse at a right angle.

Let the median be extended from the right corner to the middle of the hypotenuse, and h is the height. In this case, it turns out that:

• sin α = b / (2 * s); sin β = a / (2 * s).
• cos α = a / (2 * s); cos β = b / (2 * s).
• sin α = h / b; sin β = h / a.

Two pages

If the lengths of the hypotenuse and one of the legs are known in a right-angled triangle or on both 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 from its side.

Perimeter of an Equal Triangle can be found by concatenating the side lengths sequentially, or by multiplying the side length by 2 and adding the base length 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 is the base.

Perimeter of an equilateral triangle can be found by sequentially concatenating the length of its sides or by multiplying the length of any page by 3.

The general formula for finding the rim of equilateral triangles will 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 the 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 of the parallelogram range.

Since the area of ​​the parallelogram is the same as the product of its base height, the area of ​​the triangle will be half that product. Thus, for ΔABC, the region will be the same

Now consider a right-angled triangle:

Two identical right-angled triangles can be bent into a rectangle if it leans against them, which is each 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-angled triangle is equal to the product of the legs, divided by 2.

From these examples, it can be inferred that the surface of each triangle is the same as the product of the length, and the height is reduced to a substrate 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.

Using a calculator, extract the square root of the difference between the hypotenuse squared and the known leg, also squared. The leg is called the side of a right triangle adjacent to a right angle. This expression is derived from the Pythagorean theorem, which states that the square of the hypotenuse of a triangle is equal to the sum of the squares of the legs.

Before we look at the different ways of finding a leg in a right-angled triangle, let's take some notation. Check which of the listed cases corresponds to the condition of your problem and, depending on this, follow the corresponding paragraph. Find out what quantities in the triangle in question you know. Use the following expression to calculate the leg: a = sqrt (c ^ 2-b ^ 2), if you know the values ​​of the hypotenuse and the other leg.

The relationships between the sides and angles of this geometric figure are discussed in detail in the mathematical discipline of trigonometry. To apply this equation, you need to know the length of any two sides of a right triangle.

Calculate the length of one of the legs if the dimensions of the hypotenuse and the other leg are known. If the problem contains the hypotenuse and one of the adjacent sharp corners, use the Bradis tables.

The inner triangle will be similar to the outer one, since the midlines are parallel to the legs and hypotenuse, and are equal to their halves, respectively. Since the hypotenuse is unknown, to find the midline M_c, you need to substitute the radical from the Pythagorean theorem.

The hypotenuse is the longest side of a right triangle. It lies opposite a right angle. The length of the hypotenuse can be found in various ways. If the length of both legs is known, then its size is calculated according to the Pythagorean theorem: the sum of the squares of two legs is equal to the square of the hypotenuse. Knowing that the sum of all the angles is 180 °, we subtract the right angle and the already known one.

When calculating the parameters of a right-angled triangle, it is important to pay attention to the known values ​​and solve the problem using the simplest formula. First, let's remember what a right triangle is. A right-angled triangle is a geometric figure of three line segments that connect points that do not lie on one straight line, and one of the corners of this figure is 90 degrees. There are several ways to find out the length of the leg.

Formula: c² = a² + b², where c - hypotenuse, a and b - legs

If we know the hypotenuse and leg, then we can find the length of the unknown leg using the Pythagorean theorem. It sounds like this: "The square of the hypotenuse is equal to the sum of the squares of the legs." There are four options for finding a leg using trigonometric functions: sine, cosine, tangent, cotangent. The sine of the angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin = a / c, where a is the leg opposite a given angle, and c is the hypotenuse.

The unusual properties of right-angled triangles were discovered by the ancient Greek scientist Pythagoras, who discovered that the square of the hypotenuse in such triangles is equal to the sum of the squares of the legs

Height - perpendicular going from any vertex of the triangle, to the opposite side (or its continuation, for a triangle with an obtuse angle). The heights of the triangle intersect at one point, which is called the orthocenter. If it is an arbitrary right-angled triangle, then there is not enough data.

Also, it is useful to know the values ​​of trigonometric functions for the most typical angles of 30, 45, 60, 90, 180 degrees. If, according to the conditions, the sizes of the legs are specified, find the length of the hypotenuse. In life, we often have to deal with math problems: at school, at the university, and then helping our child with homework.

Next, we transform the formula and get: a = sin * c

To solve problems, the table below will help us. Let's consider these options. An interesting special case is when one of the acute angles is equal to 30 degrees.

People in certain professions will encounter mathematics on a daily basis.

You can also find an unknown leg if any other side and any acute angle of a right triangle are known. Find the side of a right triangle using the Pythagorean theorem. Also, the sides of a right-angled triangle can be found using various formulas depending on the number of known variables.

A rectangular triangle contains a huge variety of dependencies. This makes it an attractive object for all sorts of geometric tasks. Finding the hypotenuse is considered one of the most common tasks.

Right triangle

A right-angled triangle is a triangle that contains a right angle, i.e. an angle of 90 degrees. Only in a right-angled triangle can trigonometric functions be expressed in terms of the values ​​of the sides. Additional constructions will have to be made in an arbitrary triangle.
In a right-angled triangle, two of the three heights coincide with the sides are called legs. The third party is called the hypotenuse. The height drawn to the hypotenuse is the only one in this type of triangle, requiring additional constructions.

Rice. 1. Types of triangles.

There can be no obtuse angles in a right-angled triangle. Just as the existence of a second right angle is impossible. In this case, the identity of the sum of the angles of the triangle is violated, which is always equal to 180 degrees.

Hypotenuse

Let's go directly to the hypotenuse of the triangle. The hypotenuse is the largest side of the triangle. The hypotenuse is always greater than any of the legs, but at the same time it is always less than the sum of the legs. This is a consequence of the triangle inequality theorem.

The theorem says: in a triangle, none of the sides can be greater than the sum of the other two. There is also a second formulation or second part of the theorem: in a triangle opposite the larger side there is a larger angle and vice versa.

Rice. 2. Right-angled triangle.

In a right-angled triangle, a large angle is a right angle, since there cannot be a second right angle or obtuse angle for the reasons already mentioned. This means that there is always a large side opposite the right angle.

It seems incomprehensible why exactly a right-angled triangle has earned a separate name for each of the sides. In fact, in an isosceles triangle, the sides also carry their names: sides and base. But it is precisely for the legs and hypotenuses that teachers especially like to put deuces. Why? On the one hand, this is a tribute to the memory of the ancient Greeks, the inventors of mathematics. It was they who studied right-angled triangles and, along with this knowledge, left a whole layer of information on which modern science is based. On the other hand, the existence of these names greatly simplifies the formulation of theorems and trigonometric identities.

Pythagorean theorem

If a teacher asks about the formula for the hypotenuse of a right-angled triangle, then, with a probability of 90%, he means the Pythagorean theorem. The theorem says: in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

Rice. 3. Hypotenuse of a right-angled triangle.

Pay attention to how clearly and succinctly the theorem is formulated. Such simplicity cannot be achieved without using the concepts of hypotenuse and leg.

The theorem has the following formula:

$ c ^ 2 = b ^ 2 + a ^ 2 $ - where c is the hypotenuse, a and b are the legs of a right triangle.

What have we learned?

We talked about what a right triangle is. We found out why they came up with the names of the legs and hypotenuse at all. We found out some properties of the hypotenuse and gave the formula for the length of the hypotenuse of a triangle through the Pythagorean theorem.

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In life, we often have to deal with math problems: at school, at the university, and then helping our child with homework. People in certain professions will be exposed to mathematics on a daily basis. Therefore, it is useful to memorize or recall mathematical rules. In this article we will analyze one of them: finding the leg of a right-angled triangle.

What is a right triangle

First, let's remember what a right triangle is. A right-angled triangle is a geometric figure of three line segments that connect points that do not lie on one straight line, and one of the corners of this figure is 90 degrees. The sides forming a right angle are called legs, and the side that lies opposite the right angle is called the hypotenuse.

Find the leg of a right triangle

There are several ways to find out the length of the leg. I would like to consider them in more detail.

Pythagorean theorem to find the leg of a right triangle

If we know the hypotenuse and leg, then we can find the length of the unknown leg using the Pythagorean theorem. It sounds like this: "The square of the hypotenuse is equal to the sum of the squares of the legs." Formula: c² = a² + b², where c - hypotenuse, a and b - legs. We transform the formula and get: a² = c²-b².

Example. The hypotenuse is 5 cm, and the leg is 3 cm. We transform the formula: c² = a² + b² → a² = c²-b². Then we decide: a² = 5²-3²; a² = 25-9; a² = 16; a = √16; a = 4 (cm).

Trigonometric ratios to find the leg of a right triangle

You can also find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding a leg using trigonometric functions: sine, cosine, tangent, cotangent. To solve problems, the table below will help us. Let's consider these options.

Find the leg of a right triangle using sine

The sine of the angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin = a / c, where a is the leg opposite a given angle, and c is the hypotenuse. Next, we transform the formula and get: a = sin * c.

Example. The hypotenuse is 10 cm, the angle A is 30 degrees. According to the table, we calculate the sine of angle A, it is 1/2. Then, using the transformed formula, we solve: a = sin∠А * c; a = 1/2 * 10; a = 5 (cm).

Find the leg of a right triangle using the cosine

The cosine of the angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos = b / c, where b is the leg adjacent to the given angle, and c is the hypotenuse. Let's transform the formula and get: b = cos * c.

Example. Angle A is 60 degrees, the hypotenuse is 10 cm. According to the table, we calculate the cosine of angle A, it is 1/2. Then we decide: b = cos∠A * c; b = 1/2 * 10, b = 5 (cm).

Find the leg of a right triangle using the tangent

The tangent of the angle (tg) is the ratio of the opposite leg to the adjacent leg. Formula: tg = a / b, where a is the leg opposite to the corner, and b is adjacent. We transform the formula and get: a = tg * b.

Example. Angle A is equal to 45 degrees, hypotenuse is equal to 10 cm. According to the table we calculate the tangent of angle A, it is equal to Solve: a = tg∠A * b; a = 1 * 10; a = 10 (cm).

Find the leg of a right triangle using the cotangent

The cotangent of the angle (ctg) is the ratio of the adjacent leg to the opposite leg. Formula: ctg = b / a, where b is the leg adjacent to the corner, a is the opposite leg. In other words, a cotangent is an “inverted tangent”. We get: b = ctg * a.

Example. Angle A is 30 degrees, the opposite leg is 5 cm. According to the table, the tangent of angle A is √3. Calculate: b = ctg∠A * a; b = √3 * 5; b = 5√3 (cm).

So, now you know how to find a leg in a right triangle. As you can see, this is not so difficult, the main thing is to remember the formulas.