The two sides of a right triangle that form a right angle are called legs. The longest side of a triangle opposite the right angle is called the hypotenuse. In order to detect the hypotenuse, you need to know the length of the legs.

Instructions

1. The lengths of the legs and hypotenuse are related by a relationship that is described by the Pythagorean theorem. Algebraic formulation: “In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.” The Pythagorean formula looks like this: c2 = a2 + b2, where c is the length of the hypotenuse, a and b are the lengths of the legs.

2. Knowing the lengths of the legs, according to the Pythagorean theorem, it is possible to find the hypotenuse of a right triangle: c = ?(a2 + b2).

3. Example. The length of one of the legs is 3 cm, the length of the other is 4 cm. The sum of their squares is 25 cm?: 9 cm? + 16 cm? = 25 cm?.The length of the hypotenuse in our case is equal to the square root of 25 cm? – 5 cm. Therefore, the length of the hypotenuse is 5 cm.

The hypotenuse is the side in a right triangle that is opposite the 90 degree angle. In order to calculate its length, it is enough to know the length of one of the legs and the size of one of the acute angles of the triangle.

Instructions

1. With the famous leg and acute angle of a right triangle, the size of the hypotenuse can be equal to the ratio of the leg to the cosine/sine of this angle, if this angle is opposite/adjacent to it: h = C1 (or C2)/sin?; h = C1 (or C2 )/cos?.Example: Let a right triangle ABC with a hypotenuse AB and a right angle C be given. Let angle B be 60 degrees and angle A 30 degrees. The length of leg BC is 8 cm. We need to find the length of the hypotenuse AB. To do this, you can use any of the methods proposed above: AB = BC/cos60 = 8 cm. AB = BC/sin30 = 8 cm.

The hypotenuse is the longest side of a rectangular triangle. It is located opposite the right angle. Method for finding the hypotenuse of a rectangular triangle depends on what initial data you have.

Instructions

1. If we have rectangular legs triangle, then the length of the hypotenuse of the rectangular triangle can be discovered with the help of the Pythagorean theorem - 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 lengths of the legs of a rectangular triangle .

2. If we draw one of the legs and an acute angle, then the formula for finding the hypotenuse will depend on which angle in relation to the driven leg - adjacent (located near the leg) or opposite (located opposite it. In the case of an adjacent angle, the hypotenuse is equal to the ratio of the leg by the cosine of this angle: c = a/cos?; E is the opposite angle, the hypotenuse is equal to the ratio of the leg to the sine of the angle: c = a/sin?.

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The hypotenuse is the side of a right triangle that lies opposite the right angle. It is the longest side of a right triangle. It can be calculated using the Pythagorean theorem or using the formulas of trigonometric functions.

Instructions

1. The sides of a right triangle that are adjacent to a right angle are called legs. In the figure, the legs are designated AB and BC. Let the lengths of both legs be given. Let us denote them as |AB| and |BC|. In order to find the length of the hypotenuse |AC|, we use the Pythagorean theorem. According to this theorem, the sum of the squares of the legs is equal to the square of the hypotenuse, i.e. in the notation of our figure |AB|^2 + |BC|^2 = |AC|^2. From the formula we find that the length of the hypotenuse AC is found as |AC| = ?(|AB|^2 + |BC|^2) .

2. Let's look at an example. Let the lengths of the legs |AB| be given. = 13, |BC| = 21. By the Pythagorean theorem we find that |AC|^2 = 13^2 + 21^2 = 169 + 441 = 610. In order to obtain the length of the hypotenuse, you need to take the square root of the sum of the squares of the legs, i.e. from number 610: |AC| =?610. Using the table of squares of integers, we find out that the number 610 is not a perfect square of any integer. In order to obtain the final value of the length of the hypotenuse, let's try to move the full square from under the root sign. To do this, let's factorize the number 610. 610 = 2 * 5 * 61. Looking at the table of primitive numbers, we see that 61 is a primitive number. Consequently, the subsequent reduction of the number?610 is unrealistic. We get the final result |AC| = ?610. If the square of the hypotenuse was equal to, for example, 675, then?675 = ?(3 * 25 * 9) = 5 * 3 * ?3 = 15 * ?3. If a similar reduction is acceptable, perform a reverse check - square the total and compare it with the initial value.

3. Let us know one of the legs and the angle adjacent to it. To be specific, let these be the side |AB| and angle?. Then we can use the formula for the trigonometric function cosine - the cosine of an angle is equal to the ratio of the adjacent leg to the hypotenuse. Those. in our notation cos ? = |AB| / |AC|. From there we get the length of the hypotenuse |AC| = |AB| / cos ?.If we are familiar with the side |BC| and angle?, then we will use the formula to calculate the sine of an angle - the sine of an angle is equal to the ratio of the opposite side to the hypotenuse: sin? = |BC| / |AC|. We find that the length of the hypotenuse is |AC| = |BC| /cos?.

4. For clarity, let's look at an example. Let the length of the leg |AB| be given. = 15. And the angle? = 60°. We get |AC| = 15 / cos 60° = 15 / 0.5 = 30. Let's look at how you can check your result using the Pythagorean theorem. To do this, we need to calculate the length of the second leg |BC|. Using the formula for the tangent of the angle tg? = |BC| / |AC|, we get |BC| = |AB| *tg? = 15 * tan 60° = 15 * ?3. Next we apply the Pythagorean theorem, we get 15^2 + (15 * ?3)^2 = 30^2 => 225 + 675 = 900. The check is completed.

Helpful advice
After calculating the hypotenuse, check whether the resulting value satisfies the Pythagorean theorem.

Geometry is not a simple science. It can be useful both for the school curriculum and in real life. Knowledge of many formulas and theorems will simplify geometric calculations. One of the simplest figures in geometry is a triangle. One of the varieties of triangles, equilateral, has its own characteristics.

Features of an equilateral triangle

By definition, a triangle is a polyhedron that has three angles and three sides. This is a flat two-dimensional figure, its properties are studied in high school. Based on the type of angle, there are acute, obtuse and right triangles. A right triangle is a geometric figure where one of the angles is 90º. Such a triangle has two legs (they create a right angle) and one hypotenuse (it is opposite the right angle). Depending on what quantities are known, there are three simple ways to calculate the hypotenuse of a right triangle.

The first way is to find the hypotenuse of a right triangle. Pythagorean theorem

The Pythagorean theorem is the oldest way to calculate any of the sides of a right triangle. It sounds like this: “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.” Thus, to calculate the hypotenuse, one must derive the square root of the sum of two legs squared. For clarity, formulas and a diagram are given.

Second way. Calculation of the hypotenuse using 2 known quantities: leg and adjacent angle

One of the properties of a right triangle states that the ratio of the length of the leg to the length of the hypotenuse is equivalent to the cosine of the angle between this leg and the hypotenuse. Let's call the angle known to us α. Now, thanks to the well-known definition, you can easily formulate a formula for calculating the hypotenuse: Hypotenuse = leg/cos(α)

Third way. Calculation of the hypotenuse using 2 known quantities: leg and opposite angle

If the opposite angle is known, it is possible to again use the properties of a right triangle. The ratio of the length of the leg and the hypotenuse is equivalent to the sine of the opposite angle. Let us again call the known angle α. Now for the calculations we will use a slightly different formula:
Hypotenuse = leg/sin (α)

Examples to help you understand formulas

For a deeper understanding of each of the formulas, you should consider illustrative examples. So, suppose you are given a right triangle, where there is the following data:

• Leg – 8 cm.
• The adjacent angle cosα1 is 0.8.
• The opposite angle sinα2 is 0.8.

According to the Pythagorean theorem: Hypotenuse = square root of (36+64) = 10 cm.
According to the size of the leg and adjacent angle: 8/0.8 = 10 cm.
According to the size of the leg and the opposite angle: 8/0.8 = 10 cm.

Once you understand the formula, you can easily calculate the hypotenuse with any data.

Video: Pythagorean Theorem

There are many types of triangles: positive, isosceles, acute, and so on. All of them have properties that are classical only for them, and each has its own rules for finding quantities, be it a side or an angle at the base. But from each variety of these geometric figures, it is possible to single out a triangle with a right angle into a separate group.

You will need

• Blank sheet, pencil and ruler for a schematic representation of a triangle.

Instructions

1. A triangle is called rectangular if one of its angles is 90 degrees. It consists of 2 legs and a hypotenuse. The hypotenuse is the largest side of this triangle. It lies contrary to the right angle. The legs, accordingly, are called its smaller sides. They can be either equal to each other or have different sizes. Equality of the legs means that you are working with an isosceles right triangle. Its beauty is that it combines the properties of two figures: a right triangle and an isosceles triangle. If the legs are not equal, then the triangle is arbitrary and obeys the basic law: the larger the angle, the larger the one lying opposite it rolls.

2. There are several methods for finding the hypotenuse by leg and angle. But before using one of them, you should determine which leg and angle are known. If an angle and a leg adjacent to it are given, then the hypotenuse is easier to detect by looking at the cosine of the angle. The cosine of an acute angle (cos a) in a right triangle is the ratio of the adjacent leg to the hypotenuse. It follows that the hypotenuse (c) will be equal to the ratio of the adjacent leg (b) to the cosine of the angle a (cos a). This can be written like this: cos a=b/c => c=b/cos a.

3. If an angle and an opposite leg are given, then you should work with the sine. The sine of an acute angle (sin a) in a right triangle is the ratio of the opposite side (a) to the hypotenuse (c). The thesis here works as in the previous example, only instead of the cosine function, the sine is taken. sin a=a/c => c=a/sin a.

4. You can also use a trigonometric function such as tangent. But finding the desired value will become slightly more difficult. The tangent of an acute angle (tg a) in a right triangle is the ratio of the opposite leg (a) to the adjacent leg (b). Having discovered both legs, apply the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs) and the huge side of the triangle will be discovered.

The hypotenuse is the side in a right triangle that is opposite the 90 degree angle. In order to calculate its length, it is enough to know the length of one of the legs and the size of one of the acute angles of the triangle.

Instructions

1. With a leading leg and an acute angle of a right triangle, the size of the hypotenuse can be equal to the ratio of the leg to the cosine/sine of this angle, if this angle is opposite/adjacent to it: h = C1 (or C2)/sin?; h = C1 (or C2 )/cos?.Example: Let a right triangle ABC with a hypotenuse AB and a right angle C be given. Let angle B be 60 degrees and angle A 30 degrees. The length of leg BC is 8 cm. We need to find the length of the hypotenuse AB. To do this, you can use any of the methods proposed above: AB = BC/cos60 = 8 cm. AB = BC/sin30 = 8 cm.

Word " leg“comes from the Greek words “perpendicular” or “plumb” - this explains why both sides of a right triangle, constituting its ninety-degree angle, were named this way. Find the length of each leg It’s not difficult if you know the value of the angle adjacent to it and some other parameter, because in this case the values ​​of all 3 angles will actually become known.

Instructions

1. If, in addition to the value of the adjacent angle (β), the length of the second leg a (b), then the length leg and (a) can be defined as the quotient of the length of the famous leg and for the tangent of the desired angle: a=b/tg(β). This follows from the definition of this trigonometric function. You can do without the tangent if you use the theorem of sines. It follows from it that the ratio of the length of the desired side to the sine of the opposite angle is equal to the ratio of the length of the desired one leg and to the sine of the famous angle. Opposite to what is desired leg y acute angle can be expressed through the famous angle as 180°-90°-β = 90°-β, because the sum of all angles of any triangle must be 180°, and by the definition of a right triangle, one of its angles is equal to 90°. This means the desired length leg and can be calculated using the formula a=sin(90°-β)∗b/sin(β).

2. If the value of the adjacent angle (β) and the length of the hypotenuse (c) are known, then the length leg and (a) can be calculated as the product of the length of the hypotenuse and the cosine of the famous angle: a=c∗cos(β). This follows from the definition of cosine as a trigonometric function. But you can use, as in the previous step, the theorem of sines and then the length of the desired leg a will be equal to the product of the sine of the difference between 90° and the reference angle and the ratio of the length of the hypotenuse to the sine of the right angle. And since the sine of 90° is equal to one, the formula can be written as follows: a=sin(90°-β)∗c.

3. The actual calculations can be made, say, using the software calculator included in the Windows OS. To launch it, you can select the “Run” item in the main menu on the “Start” button, type the calc command and click the “OK” button. In the simplest version of the interface of this program that opens by default, trigonometric functions are not provided; therefore, after launching it, you need to click the “View” section in the menu and select the line “Scientist” or “Engineer” (depending on the version of the operating system used).

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The word “kathet” came into Russian from Greek. In exact translation, it means a plumb line, that is, perpendicular to the surface of the earth. In mathematics, legs are the sides that form a right angle of a right triangle. The side opposite this angle is called the hypotenuse. The term “leg” is also used in architecture and special welding technology.

Draw a right triangle DIA. Label its legs as a and b, and its hypotenuse as c. All sides and angles of a right triangle are interconnected by certain relationships. The ratio of the leg opposite one of the acute angles to the hypotenuse is called the sine of this angle. In this triangle sinCAB=a/c. Cosine is the ratio to the hypotenuse of the adjacent leg, that is, cosCAB=b/c. The inverse relationships are called secant and cosecant. The secant of a given angle is obtained by dividing the hypotenuse by the adjacent leg, that is, secCAB = c/b. The result is the reciprocal of the cosine, that is, it can be expressed using the formula secCAB=1/cosSAB. The cosecant is equal to the quotient of the hypotenuse divided by the opposite side and is the reciprocal of the sine. It can be calculated using the formula cosecCAB = 1/sinCAB Both legs are related to each other by tangent and cotangent. In this case, the tangent will be the ratio of side a to side b, that is, the opposite side to the adjacent side. This relationship can be expressed by the formula tgCAB=a/b. Accordingly, the inverse ratio will be the cotangent: ctgCAB=b/a. The relationship between the sizes of the hypotenuse and both legs was determined by the ancient Greek mathematician Pythagoras. The theorem named after him is still used by people to this day. It says that the square of the hypotenuse is equal to the sum of the squares of the legs, that is, c2 = a2 + b2. Accordingly, each leg will be equal to the square root of the difference between the squares of the hypotenuse and the other leg. This formula can be written as b=?(c2-a2). The length of the leg can also be expressed through the well-known relations. According to the theorems of sines and cosines, a leg is equal to the product of the hypotenuse and one of these functions. It can also be expressed through tangent or cotangent. Leg a can be found, say, using the formula a = b*tan CAB. In the same way, depending on the given tangent or cotangent, the 2nd leg is determined. The term “leg” is also used in architecture. It is used in relation to an Ionic capital and denotes a plumb line through the middle of its back. That is, in this case, this term denotes a perpendicular to a given line. In special welding technology there is the concept of “fillet weld leg”. As in other cases, this is the shortest distance. Here we are talking about the interval between one of the parts being welded to the boundary of the seam located on the surface of another part.

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Note!
When working with the Pythagorean theorem, remember that you are dealing with a degree. Having discovered the sum of the squares of the legs, to obtain the final result, you must extract the square root.

There are three options for solving this problem. The first is if in the conditions of the problem it is given that the legs are equal (in fact, we have a right isosceles triangle). The second is if some angle is still given (except for the 45% angle, then we have the same isosceles triangle and return to the first option). And the third - when one of the legs is known. Let's consider these options in more detail.

How to find equal legs with a known hypotenuse

• the first leg (let's denote it with the letter "a") is equal to the second leg ((let's denote it with the letter "b"): a=b;

• leg size;

In this version, the solution to the problem is based on the use of the Pythagorean theorem. It is applied to right triangles and its main version sounds like: “The square of the hypotenuse is equal to the sum of the squares of the legs.” Since our legs are equal, we can denote both legs with the same symbol: a=b, which means a=a.

1. We substitute our symbols into the theorem (taking into account the above):
   c^2=a^2+a^2,
2. Next, we simplify the formula as much as possible:
   с^2=2*(a^2) - group,
   с=√2*а - we bring both sides of the equation to the square root,
   a=c/√2 - we take out what we are looking for.
3. We substitute this value of the hypotenuse and get the solution:
   a=x/√2

How to find legs, given a known hypotenuse and angle

• the hypotenuse (let's denote it by the letter "c") is equal to x cm: c=x;
• angle β equal to q: β=q;

• leg size;

To solve this problem you need to use trigonometric functions. The most popular two of them are:

• sine function - the sine of the desired angle is equal to the ratio of the opposite side to the hypotenuse;
• cosine function - the cosine of the desired angle is equal to the ratio of the adjacent leg to the hypotenuse;

You can use any one. I'll give an example using the first one. Let the legs be designated by the symbols “a” (adjacent to the corner) and “b” (opposite to the corner). Accordingly, our angle lies between leg “a” and the hypotenuse.

1. We substitute the selected symbols into the formula:
   sinβ = b/c
2. We derive the leg:
   b=c*sinβ
3. We substitute our given and we have one leg.
   b=c*sinq

The second leg can be found using the second trigonometric function, or go to the third option.

How to find one side if the hypotenuse and the other side are known

• the hypotenuse (let's denote it by the letter "c") is equal to x cm: c=x;
• leg (let's denote it by the letter "b") is equal to y cm: b=y;

• the size of the other leg (let’s denote it by the letter “a”);

In this version, the solution to the problem, as in the first, is to use the Pythagorean theorem.

1. We substitute our symbols into the theorem:
   c^2=a^2+b^2,
2. We take out the necessary leg:
   a^2=c^2-b^2
3. We take both sides of the equation to the square root:
   a=√(c^2-b^2)
4. We substitute these values ​​and we have the solution:
   a=√(x^2-y^2)

Instructions

Let one of the legs of a right triangle be known. Suppose |BC| = b. Then we can use the Pythagorean theorem, according to the hypotenuse is equal to the sum of the squares of the legs: a^2 + b^2 = c^2. From this equation we find the unknown side |AB| = a = √ (c^2 - b^2).

Let one of the angles of a right triangle be known, suppose ∟α. Then AB and BC of right triangle ABC can be found using trigonometric functions. So we get: sine ∟α is equal to the ratio of the opposite side sin α = b / c, cosine ∟α is equal to the ratio of the adjacent side to the hypotenuse cos α = a / c. From here we find the required side lengths: |AB| = a = c * cos α, |BC| = b = c * sin α.

Let the ratio of the legs k = a / b be known. We also solve the problem using trigonometric functions. The ratio a / b is nothing more than the cotangent ∟α: the adjacent side ctg α = a / b. In this case, from this equality we express a = b * ctg α. And we substitute a^2 + b^2 = c^2 into the Pythagorean theorem:

b^2 * cotg^2 α + b^2 = c^2. Taking b^2 out of brackets, we get b^2 * (ctg^2 α + 1) = c^2. And from here we easily obtain the length of the leg b = c / √(ctg^2 α + 1) = c / √(k^2 + 1), where k is the given ratio of the legs.

By analogy, if the ratio of the legs b / a is known, we solve the problem using the tangent tan α = b / a. We substitute the value b = a * tan α into the Pythagorean theorem a^2 * tan^2 α + a^2 = c^2. Hence a = c / √(tg^2 α + 1) = c / √(k^2 + 1), where k is the given ratio of the legs.

Let's consider special cases.

∟α = 30°. Then |AB| = a = c * cos α = c * √3 / 2; |BC| = b = c * sin α = c / 2.

∟α = 45°. Then |AB| = |BC| = a = b = c * √2 / 2.

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note

Square roots are extracted with a positive sign, because length cannot be negative. This seems obvious, but this error is very common if you solve the problem automatically.

Helpful advice

To find the legs of a right triangle, it is convenient to use the reduction formulas: sin β = sin (90° - α) = cos α; cos β = cos (90° - α) = sin α.

Sources:

• Bradis tables for finding values ​​of trigonometric functions

The relationships between the sides and angles of a right triangle are discussed in the branch of mathematics called trigonometry. To find the sides of a right triangle, it is enough to know the Pythagorean theorem, the definitions of trigonometric functions, and have some means for finding the values ​​of trigonometric functions, for example, a calculator or Bradis tables. Let us consider below the main cases of problems of finding the sides of a right triangle.

You will need

• Calculator, Bradis tables.

Instructions

If you are given one of the acute angles, for example, A, and the hypotenuse, then the legs can be found from the definitions of the basic trigonometric ones:

a= c*sin(A), b= c*cos(A).

If one of the acute angles, for example, A, and one of the legs, for example, a, is given, then the hypotenuse and the other leg are calculated from the relations: b=a*tg(A), c=a*sin(A).

Helpful advice

If you do not know the value of the sine or cosine of one of the angles necessary for calculation, you can use the Bradis tables; they provide the values ​​of trigonometric functions for a large number of angles. In addition, most modern calculators are capable of calculating sines and cosines of angles.

Sources:

• how to calculate the side of a right triangle in 2019

Tip 3: How to find an angle if you know the sides of a right triangle

Tre square, one of the angles of which is right (equal to 90°) is called rectangular. Its longest side always lies opposite the right angle and is called the hypotenuse, and the other two sides are called legs. If the lengths of these three sides are known, then find the values ​​of all angles of three square and will not be difficult, since in fact you only need to calculate one of the angles. There are several ways to do this.

Instructions

Use to calculate the quantities (α, β, γ) the definitions of trigonometric functions through a rectangular triangle. Such, for example, for the sine of an acute angle as the ratio of the length of the opposite leg to the length of the hypotenuse. This means that if the lengths of the legs (A and B) and the hypotenuse (C), then, for example, you can find the sine of the angle α lying opposite leg A by dividing the length sides And for the length sides C (hypotenuse): sin(α)=A/C. Having found out the value of the sine of this angle, you can find its value in degrees using the inverse function of the sine - arcsine. That is, α=arcsin(sin(α))=arcsin(A/C). In the same way you can find the size of an acute angle in a triangle. square Yes, but this is not necessary. Since the sum of all angles is three square a is 180°, and in three square If one of the angles is 90°, then the value of the third angle can be calculated as the difference between 90° and the value of the found angle: β=180°-90°-α=90°-α.

Instead of defining the sine, you can use the definition of the cosine of an acute angle, which is formulated as the ratio of the length of the leg adjacent to the desired angle to the length of the hypotenuse: cos(α)=B/C. Here again, use the inverse trigonometric function (arccosine) to find the angle in degrees: α=arccos(cos(α))=arccos(B/C). After this, as in the previous step, all that remains is to find the value of the missing angle: β=90°-α.

You can use a similar tangent - it is expressed by the ratio of the length of the leg opposite the desired angle to the length of the adjacent leg: tan(α)=A/B. Again, determine the angle in degrees using the inverse trigonometric function -: α=arctg(tg(α))=arctg(A/B). The formula for the missing angle will remain unchanged: β=90°-α.

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Tip 4: How to find the side length of a right triangle

A triangle is considered to be right-angled if one of its angles is right. Side triangle located opposite the right angle is called the hypotenuse, and the other two sides- legs. To find the lengths of the sides of a rectangular triangle, you can use several methods.

Instructions

You can find out the third sides, knowing the lengths of the other two sides triangle. This can be done using the Pythagorean theorem, which states that a square of a rectangular triangle the sum of the squares of its legs. (a² = b²+ c²). From here we can express the lengths of all sides of a rectangular triangle:
b² = a² - c²;
c² = a² - b²
For example, for a rectangular triangle the length of the hypotenuse a (18 cm) and one of the legs, for example c (14 cm), is known. To length another side, you need to perform 2 algebraic operations:
c² = 18² - 14² = 324 - 196 = 128 cm
c = √128 cm
Answer: leg length is √128 cm or approximately 11.3 cm

You can resort to if you know the length of the hypotenuse and the size of one of the acute points of a given rectangular triangle. Let the length be c and one of the acute angles be equal to α. In this case, find 2 others sides rectangular triangle it will be possible using the following formulas:
a = с*sinα;
b = с*cosα.
You can give: the length of the hypotenuse is 15 cm, one of the acute angles is 30 degrees. To find the lengths of the other two sides you need to perform 2 steps:
a = 15*sin30 = 15*0.5 = 7.5 cm
b = 15*cos30 = (15*√3)/2 = 13 cm (approx.)

The most non-trivial way to find length sides rectangular triangle- is to express it from the perimeter of a given figure:
P = a + b + c, where P is the perimeter of the rectangular triangle. From this expression it is easy to express length any side of a rectangular triangle.

Tip 5: How to find the angle of a right triangle knowing all the sides

Knowledge of all three sides directly coal triangle is more than enough to calculate any of its angles. There is so much information that you even have the opportunity to choose which parties to use in the calculations in order to use the trigonometric function that suits you best.

Instructions

If you prefer to deal with the arcsine, use the length of the hypotenuse (C) - the longest sides- and that leg (A) that lies opposite the desired angle (α). Dividing the length of this leg by the length of the hypotenuse will give the value of the sine of the desired angle, and the inverse function of the sine - the arcsine - from the resulting value will restore the value of the angle in . Therefore, use the following in your calculations: α = arcsin(A/C).

To replace arcsine with arccosine, use the length calculations of those sides that form the desired angle (α). One of them will be the hypotenuse (C), and the other will be the leg (B). By definition, the cosine is the length of the leg adjacent to the angle to the length of the hypotenuse, and the angle from the cosine value is the arc cosine function. Use the following calculation formula: α = arccos(B/C).

Can be used in calculations. To do this, you need the lengths of the two short sides - the legs. Tangent of an acute angle (α) in a straight line coal triangle is determined by the ratio of the length of the leg (A) lying opposite it to the length of the adjacent leg (B). By analogy with the options described above, use the following formula: α = arctan(A/B).

Formula

Which triangle is called a right triangle?

There are several types of triangles. Some have all acute angles, others have one obtuse and two acute, and others have two acute and one straight. On this basis, each type of these geometric shapes was named: acute-angled, obtuse-angled and rectangular. That is, a triangle in which one of the angles is 90° is called a right triangle. There is another thing similar to the first. A triangle whose two sides are perpendicular is called a right triangle.

Hypotenuse and legs

In acute and obtuse triangles, the segments connecting the vertices of the angles are simply called sides. The side also has other names. Those adjacent to the right angle are called legs. The side opposite the right angle is called the hypotenuse. Translated from Greek, the word “hypotenuse” means “tight”, and “cathetus” means “perpendicular”.

Relationships between the hypotenuse and legs

The sides of a right triangle are connected by certain relationships, which greatly facilitate calculations. For example, knowing the dimensions of the legs, you can calculate the length of the hypotenuse. This relationship, named after the person who discovered it, is called the Pythagorean theorem and it looks like this:

c2=a2+b2, where c is the hypotenuse, a and b are the legs. That is, the hypotenuse will be equal to the square root of the sum of the squares of the legs. To find any of the legs, it is enough to subtract the square of the other leg from the square of the hypotenuse and take the square root from the resulting difference.

Adjacent and opposite leg

Draw a right triangle DIA. The letter C usually denotes the vertex of a right angle, A and B - the vertices of acute angles. It is convenient to call the sides opposite each angle a, b and c, after the names of the angles opposite them. Consider angle A. Side a will be opposite for it, side b will be adjacent. The ratio of the opposite side to the hypotenuse is called. This trigonometric function can be calculated using the formula: sinA=a/c. The ratio of the adjacent leg to the hypotenuse is called cosine. It is calculated using the formula: cosA=b/c.

Thus, knowing the angle and one of the sides, you can use these formulas to calculate the other side. Both sides are also connected by trigonometric relations. The ratio of the opposite to the adjacent is called tangent, and the ratio of adjacent to the opposite is called cotangent. These relationships can be expressed by the formulas tgA=a/b or ctgA=b/a.