Geometry is not a simple science. It requires special attention and knowledge of exact formulas. This type of mathematics came to us from Ancient Greece and even after several thousand years it does not lose its relevance. Do not think in vain that this is a useless subject that bothers the heads of students and schoolchildren. In fact, geometry is applicable in many areas of life. Without knowledge of geometry, not a single architectural structure is built, cars, spaceships and airplanes are not created. Complex and not very complex road junctions and ruts - all this requires geometric calculations. Yes, even sometimes you cannot make repairs in your room without knowing basic formulas. So don't underestimate the importance of this subject. We study the most common formulas that we have to use in many solutions at school. One of them is finding the hypotenuse in a right triangle. To understand this, read below.

Before we start practicing, let's start with the basics and define what the hypotenuse is in a right triangle.

The hypotenuse is one of the sides in a right triangle that is opposite the 90 degree angle (right angle) and is always the longest.

There are several ways to find the length of the desired hypotenuse in a given right triangle.

In the case when the legs are already known to us, we use the Pythagorean theorem, where we add the sum of the squares of two legs, which will be equal to the square of the hypotenuse.

a and b are legs, c is the hypotenuse.

In our case, for a right triangle, accordingly, the formula will be as follows:

If we substitute the known numbers of legs a and b, let it be a=3 and b=4, then c=√32+42, then we get c=√25, c=5

When we know the length of only one leg, the formula can be transformed to find the length of the second. It looks like this:

In the case when, according to the conditions of the problem, we know leg A and hypotenuse C, then we can calculate the right angle of the triangle, let's call it α.

To do this we use the formula:

Let the second angle we need to calculate be β. Considering that we know the sum of the angles of a triangle, which is 180°, then: β= 180°-90°-α

In the case when we know the values of the legs, we can use the formula to find the value of the acute angle of the triangle:

Depending on the known generally accepted values, the sides of a rectangle can be found using many different formulas. Here are some of them:

When solving problems with finding unknowns in a right triangle, it is very important to focus on the values you already know and, based on this, substitute them into the desired formula. It will be difficult to remember them right away, so we advise you to make a small handwritten hint and paste it into your notebook.

As you can see, if you delve into all the intricacies of this formula, you can easily figure it out. We recommend trying to solve several problems based on this formula. After you see your result, it will become clear to you whether you understood this topic or not. Try not to memorize, but to delve into the material, it will be much more useful. Memorized material is forgotten after the first test, and you will encounter this formula quite often, so first understand it, and then memorize it. If these recommendations do not have a positive effect, then it makes sense to take additional classes on this topic. And remember: teaching is light, not teaching is darkness!

Instructions

If you need to calculate using the Pythagorean theorem, use the following algorithm: - Determine in a triangle which sides are the legs and which are the hypotenuse. The two sides forming an angle of ninety degrees are the legs, the remaining third is the hypotenuse. (cm) - Raise each leg of this triangle to the second power, that is, multiply by itself. Example 1. Suppose we need to calculate the hypotenuse if one leg in a triangle is 12 cm and the other is 5 cm. First, the squares of the legs are equal: 12 * 12 = 144 cm and 5 * 5 = 25 cm. Next, determine the sum of the squares legs. A certain number is hypotenuse, you need to get rid of the second power of the number to find length this side of the triangle. To do this, extract the value of the sum of the squares of the legs from the square root. Example 1. 144+25=169. The square root of 169 is 13. Therefore, the length of this hypotenuse equal to 13 cm.

Another way to calculate length hypotenuse lies in the terminology of sine and angles in a triangle. By definition: the sine of the angle alpha - the opposite leg to the hypotenuse. That is, looking at the figure, sin a = CB / AB. Hence, hypotenuse AB = CB / sin a. Example 2. Let the angle be 30 degrees, and the opposite side be 4 cm. We need to find the hypotenuse. Solution: AB = 4 cm / sin 30 = 4 cm / 0.5 = 8 cm. Answer: length hypotenuse equal to 8 cm.

A similar way to find hypotenuse from the definition of cosine of an angle. The cosine of an angle is the ratio of the side adjacent to it and hypotenuse. That is, cos a = AC/AB, hence AB = AC/cos a. Example 3. In triangle ABC, AB is the hypotenuse, angle BAC is 60 degrees, leg AC is 2 cm. Find AB.
Solution: AB = AC/cos 60 = 2/0.5 = 4 cm Answer: The hypotenuse is 4 cm in length.

Useful advice

When finding the value of the sine or cosine of an angle, use either the table of sines and cosines or the Bradis table.

Tip 2: How to find the length of the hypotenuse in a right triangle

The hypotenuse is the longest side in a right triangle, so it is not surprising that the word is translated from Greek as “stretched.” This side always lies opposite the 90° angle, and the sides forming this angle are called legs. Knowing the lengths of these sides and the values of the acute angles in different combinations of these values, we can calculate the length of the hypotenuse.

Instructions

If the lengths of both triangles (A and B) are known, then use the lengths of the hypotenuse (C), perhaps the most famous mathematical postulate - the Pythagorean theorem. It states that the square of the length of the hypotenuse is the sum of the squares of the lengths of the legs, from which it follows that you should calculate the root of the sum of the squared lengths of the two sides: C = √ (A² + B²). For example, if the length of one leg is 15 and - 10 centimeters, then the length of the hypotenuse will be approximately 18.0277564 centimeters, since √(15²+10²)=√(225+100)= √325≈18.0277564.

If the length of only one of the legs (A) in a right triangle is known, as well as the value of the angle opposite it (α), then the length of the hypotenuse (C) can be used using one of the trigonometric functions - the sine. To do this, divide the length of the known side by the sine of the known angle: C=A/sin(α). For example, if the length of one of the legs is 15 centimeters, and the angle at the opposite vertex of the triangle is 30°, then the length of the hypotenuse will be equal to 30 centimeters, since 15/sin(30°)=15/0.5=30.

If in a right triangle the size of one of the acute angles (α) and the length of the adjacent leg (B) are known, then to calculate the length of the hypotenuse (C) you can use another trigonometric function - cosine. You should divide the length of the known leg by the cosine of the known angle: C=B/ cos(α). For example, if the length of this leg is 15 centimeters, and the acute angle adjacent to it is 30°, then the length of the hypotenuse will be approximately 17.3205081 centimeters, since 15/cos(30°)=15/(0.5* √3)=30/√3≈17.3205081.

Length is usually used to denote the distance between two points on a line segment. It can be a straight, broken or closed line. You can calculate the length quite simply if you know some other indicators of the segment.

Among the numerous calculations performed to calculate various different quantities is finding the hypotenuse of a triangle. Recall that a triangle is a polyhedron that has three angles. Below are several ways to calculate the hypotenuse of various triangles.

First, let's look at how to find the hypotenuse of a right triangle. For those who have forgotten, a triangle with an angle of 90 degrees is called a right triangle. The side of the triangle located on the opposite side of the right angle is called the hypotenuse. In addition, it is the longest side of the triangle. Depending on the known values, the length of the hypotenuse is calculated as follows:

The lengths of the legs are known. The hypotenuse in this case is calculated using the Pythagorean theorem, which reads as follows: the square of the hypotenuse is equal to the sum of the squares of the legs. If we consider a right triangle BKF, where BK and KF are legs, and FB is the hypotenuse, then FB2= BK2+ KF2. From the above it follows that when calculating the length of the hypotenuse, each of the values of the legs must be squared in turn. Then add the learned numbers and extract the square root from the result.

Consider an example: Given a triangle with a right angle. One leg is 3 cm, the other is 4 cm. Find the hypotenuse. The solution looks like this.

FB2= BK2+ KF2= (3cm)2+(4cm)2= 9cm2+16cm2=25cm2. Extract and get FB=5cm.

The leg (BK) and the angle adjacent to it, which is formed by the hypotenuse and this leg, are known. How to find the hypotenuse of a triangle? Let us denote the known angle α. According to the property which states that the ratio of the length of the leg to the length of the hypotenuse is equal to the cosine of the angle between this leg and the hypotenuse. Considering a triangle, this can be written like this: FB= BK*cos(α).
The leg (KF) and the same angle α are known, only now it will be opposite. How to find the hypotenuse in this case? Let us turn to the same properties of a right triangle and find out that the ratio of the length of the leg to the length of the hypotenuse is equal to the sine of the angle opposite the leg. That is, FB= KF * sin (α).

Let's look at an example. Given the same right triangle BKF with hypotenuse FB. Let the angle F be equal to 30 degrees, the second angle B corresponds to 60 degrees. The BK leg is also known, the length of which corresponds to 8 cm. The required value can be calculated as follows:

FB = BK /cos60 = 8 cm.
FB = BK /sin30 = 8 cm.

Known (R), described around a triangle with a right angle. How to find the hypotenuse when considering such a problem? From the property of a circle circumscribed around a triangle with a right angle, it is known that the center of such a circle coincides with the point of the hypotenuse, dividing it in half. In simple words, the radius corresponds to half the hypotenuse. Hence the hypotenuse is equal to two radii. FB=2*R. If you are given a similar problem in which not the radius, but the median is known, then you should pay attention to the property of a circle circumscribed around a triangle with a right angle, which says that the radius is equal to the median drawn to the hypotenuse. Using all these properties, the problem is solved in the same way.

If the question is how to find the hypotenuse of an isosceles right triangle, then you need to turn to the same Pythagorean theorem. But, first of all, remember that an isosceles triangle is a triangle that has two identical sides. In the case of a right triangle, the sides are equal. We have FB2= BK2+ KF2, but since BK= KF we have the following: FB2=2 BK2, FB= BK√2

As you can see, knowing the Pythagorean theorem and the properties of a right triangle, solving problems in which it is necessary to calculate the length of the hypotenuse is very simple. If it is difficult to remember all the properties, learn ready-made formulas, substituting known values into which you can calculate the desired length of the hypotenuse.

A triangle is a geometric number consisting 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 conditions of the problem.

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To calculate the sides of a right triangle, the Pythagorean theorem is used, which states that the square of the hypotenuse is equal to the sum of the squares of the legs.

If we label the legs as "a" and "b" and the hypotenuse as "c", then the 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 angle opposite the base, "a" is the adjacent angle to calculate the pages can use the following formulas:

Two corners and a 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 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 information known about it.

You'll need it

Depending on the case, skills 2 three sides of the triangle, as well as one of its acute angles.

instructions

first Method 1. If all three pages are known triangle Then, regardless of whether 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 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 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 triangle will be carried out according to the formula: P = a * (1 / tg?

1/son? + 1)

fifths Method 5.

Online triangle calculation

Let our foot lead and be included in it, then the range will be calculated as: P = A * (1 / CTG + 1 / + 1 cos?)

Related videos

The Pythagorean theorem is the basis of all mathematics. Determines the relationship between the sides of a true triangle. There are now 367 proofs of this theorem.

instructions

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 resort to square the lengths 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 the hypotenuse, which is 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 equal to 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 unfounded 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 triplet, 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, 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 don't need A.

fifths The Pythagorean theorem is a special case, greater than the general cosine theorem, which establishes the 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 the side in a right triangle that is opposite the 90 degree angle.

instructions

first In the case of known catheters, as well as the 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 = C1 (or C2) / sin, H = C1 (or C2?) / 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 a rectangle triangle. It is located at a right angle. Method for finding the hypotenuse of a rectangle 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 discovered by a 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 one of the legs is known and at an acute angle, the formula for finding the hypotenuse will depend on the presence or absence at a certain angle in relation to the known leg - adjacent (the leg is located close), or vice versa (the opposite case is located nego.V of the specified angle is equal to the fraction hypotenuse of the leg in cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio of sine angles: da = a / sin.

Related videos

Useful tips
An angular triangle whose sides are related 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 Jero's triangles, in which pages and area are represented by integers.

A triangle is called a rectangle whose angle is 90°. The side opposite the right corner is called the hypotenuse, the other 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 equal.

To calculate the angle of a right congruent triangle, you need to know that:

This is 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 equal to 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 often 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 remain sharp.

Calculate triangle online

If you want to find them, you need to know that:

Other ways

The values of the acute angles of a right triangle can be calculated from the average - 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 let h be 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 triangle or on both sides, then trigonometric identities are used to determine the values of the 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 of 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 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 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 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 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 a parallelogram is the same as the product of its base height, the area of the triangle will be equal to half of this product. Thus, 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 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 triangle is equal to the product of the legs divided by 2.

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

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

What is a right triangle

First, let's remember what a right triangle is. A right triangle is a geometric figure of three segments that connect points that do not lie on the same straight line, and one of the angles 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.

Finding 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 side of a right triangle

If we know the hypotenuse and the 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 is the hypotenuse, a and b are the 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. Next we solve: 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. The table below will help us solve problems. Let's consider these options.

Find the leg of a right triangle using sine

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

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

Find the leg of a right triangle using cosine

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

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

Find the leg of a right triangle using tangent

Tangent of an angle (tg) is the ratio of the opposite side to the adjacent side. Formula: tg=a/b, where a is the side opposite to the angle, and b is the adjacent side. Let's transform the formula and get: a=tg*b.

Example. Angle A is equal to 45 degrees, the hypotenuse is equal to 10 cm. Using 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 cotangent

Angle cotangent (ctg) is the ratio of the adjacent side to the opposite side. Formula: ctg=b/a, where b is the leg adjacent to the angle, and is the opposite leg. In other words, 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. We calculate: b=ctg∠A*a; b=√3*5; b=5√3 (cm).