Before finding the hypotenuse of a triangle, you need to understand what features this figure has. Let's consider the main ones:
- In a right triangle, both acute angles add up to 90º.
- A leg lying opposite an angle of 30º will be equal to ½ the size of the hypotenuse.
- If the leg is equal to ½ of the hypotenuse, then the second angle will have the same value - 30º.
There are several ways to find the hypotenuse in a right triangle. The most simple solution is a calculation through legs. Let's say you know the values of the sides A and B. Then the Pythagorean theorem comes to the rescue, telling us that if we square each value of the side and sum up the data obtained, we will find out what the hypotenuse is equal to. So we just need to extract the square root value:
For example, if leg A = 3 cm and leg B = 4 cm, then the calculation will look like this:
How to find the hypotenuse through an angle?
Another way to find out what the hypotenuse is in a right triangle is to calculate through a given angle. To do this, we need to derive the value through the sine formula. Let's say we know the size of the leg (A) and the value of the opposite angle (α). Then the whole solution is contained in one formula: C=A/sin(α).
For example, if the leg length is 40 cm and the angle is 45°, then the length of the hypotenuse can be derived as follows:
The required value can also be determined through the cosine of a given angle. Let's say we know the value of one leg (B) and an acute adjacent angle (α). Then to solve the problem you will need one formula: C=B/ cos(α).
For example, if the leg length is 50 cm and the angle is 45°, then the hypotenuse can be calculated as follows:
Thus, we looked at the main ways to find out the hypotenuse in a triangle. When solving a problem, it is important to concentrate on the available data, then finding the unknown quantity will be quite simple. You only need to know a couple of formulas and the process of solving problems will become simple and enjoyable.
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 the 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 sharp corner? 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 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 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 the number 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.
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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 known value 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 methods
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.
The first are the segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is located opposite the angle of 90 degrees. A Pythagorean triangle is one whose sides are equal to the natural numbers; their lengths in this case are called “Pythagorean triple”.
Egyptian triangle
In order for the current generation to recognize geometry in the form in which it is taught in school now, it has developed over several centuries. The fundamental point is considered to be the Pythagorean theorem. The sides of a rectangular is known throughout the world) are 3, 4, 5.
Few people are not familiar with the phrase “ Pythagorean pants equal in all directions." However, in reality the theorem sounds like this: c 2 (square of the hypotenuse) = a 2 + b 2 (sum of squares of the legs).
Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called “Egyptian”. The interesting thing is 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 with 12 knots tied on it. In this case, the probability of constructing a right triangle increased to 95%.
Signs of equality of figures
- An acute angle in a right triangle and a long side, which are equal to the same elements in the second triangle, are an indisputable sign of equality of 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 according to the second criterion.
- When superimposing two figures on top of each other, we rotate them so that, when combined, they become one isosceles triangle. According to its property, the sides, or more precisely, the hypotenuses, are equal, as well as the angles at the base, which means that these figures are the same.
Based on the first sign, it is very easy to prove that the triangles are indeed equal, the main thing is that the two smaller sides (i.e., the legs) are equal to each other.
The triangles will be identical according to the second criterion, the essence of which is the equality of the leg and the acute angle.
Properties of a triangle with a right angle
The height that is lowered from the right angle splits the figure into two equal parts.
The sides of a right triangle and its median can be easily recognized by the rule: the median that falls on 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°, 45° and 60° apply.
- With an angle of 30°, it should be remembered that the opposite leg will be equal to 1/2 of the largest side.
- If the angle is 45°, then the second acute angle is also 45°. This suggests that the triangle is isosceles and its legs are the same.
- The property of an angle of 60° is that the third angle has a degree measure of 30°.
The area can be easily found out using one of three formulas:
- through the height and the side on which it descends;
- according to Heron's formula;
- on the sides and the angle between them.
The sides of a right triangle, or rather the legs, converge with two altitudes. 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 a relationship between twice the area and the length of the hypotenuse. The most common expression among students is the first one, as it requires fewer calculations.
Theorems applying to right triangle
Right triangle geometry involves the use of theorems such as:
After studying a topic about right triangles, students often forget all the information about them. Including how to find the hypotenuse, not to mention what it is.
And in vain. Because in the future the diagonal of the rectangle turns out to be this very hypotenuse, and it needs to be found. Or the diameter of a circle coincides with the largest side of a triangle, one of the angles of which is right. And it is impossible to find it without this knowledge.
There are several options for finding the hypotenuse of a triangle. The choice of method depends on the initial data set in the problem of quantities.
Method number 1: both sides are given
This is the most memorable method because it uses the Pythagorean theorem. Only sometimes students forget that this formula is used to find the square of the hypotenuse. This means that to find the side itself, you will need to take the square root. Therefore, the formula for the hypotenuse, which is usually denoted by the letter “c,” will look like this:
c = √ (a 2 + b 2), where the letters “a” and “b” represent both legs of a right triangle.
Method number 2: the leg and the angle adjacent to it are known
In order to learn how to find the hypotenuse, you will need to remember trigonometric functions. Namely cosine. For convenience, we will assume that leg “a” and the angle α adjacent to it are given.
Now we need to remember that the cosine of the angle of a right triangle is equal to the ratio of the two sides. The numerator will contain the value of the leg, and the denominator will contain the hypotenuse. It follows from this that the latter can be calculated using the formula:
c = a / cos α.
Method number 3: given a leg and an angle that lies opposite it
In order not to get confused in the formulas, let’s introduce the designation for this angle - β, and leave the side the same “a”. In this case, you will need another trigonometric function - sine.
As in the previous example, the sine is equal to the ratio of the leg to the hypotenuse. The formula for this method looks like this:
c = a / sin β.
In order not to get confused in trigonometric functions, you can remember a simple mnemonic: if the problem deals with pr O opposite angle, then you need to use it with And well, if - oh pr And lying down, then to O sinus. You should pay attention to the first vowels in keywords. They form pairs o-i or and about.
Method number 4: along the radius of the circumscribed circle
Now, in order to find out how to find the hypotenuse, you will need to remember the property of the circle that is circumscribed around a right triangle. It reads as follows. The center of the circle coincides with the middle of the hypotenuse. To put it another way, the longest side of a right triangle is equal to the diagonal of the circle. That is, double the radius. The formula for this problem will look like this:
c = 2 * r, where the letter r denotes the known radius.
This is all possible ways how to find the hypotenuse of a right triangle. For each specific task, you need to use the method that is most suitable for the data set.
Example task No. 1
Condition: in a right triangle, medians are drawn to both sides. The length of the one drawn to the larger side is √52. The other median has length √73. You need to calculate the hypotenuse.
Since medians are drawn in a triangle, they divide the legs into two equal segments. For convenience of reasoning and searching for how to find the hypotenuse, you need to introduce several notations. Let both halves of the larger leg be designated by the letter “x”, and the other by “y”.
Now we need to consider two right triangles whose hypotenuses are the known medians. For them you need to write the formula of the Pythagorean theorem twice:
(2y) 2 + x 2 = (√52) 2
(y) 2 + (2x) 2 = (√73) 2.
These two equations form a system with two unknowns. Having solved them, it will be easy to find the legs of the original triangle and from them its hypotenuse.
First you need to raise everything to the second power. It turns out:
4y 2 + x 2 = 52
y 2 + 4x 2 = 73.
From the second equation it is clear that y 2 = 73 - 4x 2. This expression needs to be substituted into the first one and calculated “x”:
4(73 - 4x 2) + x 2 = 52.
After conversion:
292 - 16 x 2 + x 2 = 52 or 15x 2 = 240.
From the last expression x = √16 = 4.
Now you can calculate "y":
y 2 = 73 - 4(4) 2 = 73 - 64 = 9.
According to the conditions, it turns out that the legs of the original triangle are equal to 6 and 8. This means that you can use the formula from the first method and find the hypotenuse:
√(6 2 + 8 2) = √(36 + 64) = √100 = 10.
Answer: hypotenuse equals 10.
Example task No. 2
Condition: calculate the diagonal drawn in a rectangle with a shorter side equal to 41. If it is known that it divides the angle into those that are related as 2 to 1.
In this problem, the diagonal of a rectangle is the longest side in a 90º triangle. So it all comes down to how to find the hypotenuse.
The problem is about angles. This means that you will need to use one of the formulas that contains trigonometric functions. First you need to determine the size of one of the acute angles.
Let the smaller of the angles discussed in the condition be designated α. Then the right angle that is divided by the diagonal will be equal to 3α. The mathematical notation for this looks like this:
From this equation it is easy to determine α. It will be equal to 30º. Moreover, it will lie opposite the smaller side of the rectangle. Therefore, you will need the formula described in method No. 3.
The hypotenuse is equal to the ratio of the leg to the sine of the opposite angle, that is:
41 / sin 30º = 41 / (0.5) = 82.
Answer: The hypotenuse is 82.