After studying the topic of right triangles, students often throw all the information about them out of their heads. 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 the circle coincides with the largest side of the triangle, one of the angles of which is right. And it is impossible to find it without this knowledge.
There are several ways to find the hypotenuse of a triangle. The choice of method depends on the initial data set in the problem of quantities.
Method number 1: both legs are given
This is the most memorable method because it uses the Pythagorean theorem. Only sometimes students forget that this formula is the square of the hypotenuse. So, 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 + a 2), where the letters "a" and "b" are written 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 need to remember the trigonometric functions. Namely cosine. For convenience, we will assume that the leg "a" and the angle α adjacent to it are given.
Now we need to remember that the cosine of the angle right triangle is equal to the ratio of the two sides. The numerator will be the value of the leg, and the denominator will be the hypotenuse. From this it follows that the latter can be calculated by the formula:
c = a / cos α.
Method number 3: given the leg and the angle that lies opposite it
In order not to get confused in the formulas, we introduce the designation for this angle - β, and leave the side as "a". In this case, another trigonometric function is required - the 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 \u003d a / sin β.
In order not to get confused in trigonometric functions, you can remember a simple mnemonic rule: if the problem is about O opposite corner, then you need to use with And nous if - oh pr And lying, then to O sinus. Pay attention to the first vowels in keywords. They form pairs oh and 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 need to remember the property of the circle, which is described around a right triangle. It reads as follows. The center of the circle coincides with the midpoint of the hypotenuse. In other words, the longest side of a right triangle is equal to the diagonal of the circle. That is, double the radius. The formula for this task would look like this:
c = 2 * r, where r denotes the known radius.
These are all possible ways to find the hypotenuse of a right triangle. In each specific task, you need to use the method that is more suitable for the data set.
Example of task #1
Condition: in a right-angled triangle, medians are drawn to both legs. The length of the one drawn to the larger side is √52. The other median has a length of √73. You need to calculate the hypotenuse.
Since medians are drawn in a triangle, they divide the legs into two equal segments. For the convenience of reasoning and finding how to find the hypotenuse, you need to introduce several notations. Let both halves of the larger leg be marked with the letter “x”, and the other with “y”.
Now we need to consider two right-angled triangles, the hypotenuses of which are known medians. For them, you need to write down 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 its hypotenuse from them.
First you need to raise everything to the second degree. It turns out:
4y 2 + x 2 = 52
y 2 + 4x 2 = 73.
It can be seen from the second equation that y 2 \u003d 73 - 4x 2. This expression must be substituted into the first and calculate "x":
4 (73 - 4x 2) + x 2 \u003d 52.
After conversion:
292 - 16 x 2 + x 2 \u003d 52 or 15 x 2 \u003d 240.
From the last expression x = √16 = 4.
Now you can calculate "y":
y 2 \u003d 73 - 4 (4) 2 \u003d 73 - 64 \u003d 9.
According to the condition, it turns out that the legs of the original triangle are 6 and 8. So, you can use the formula from the first method and find the hypotenuse:
√(6 2 + 8 2) = √(36 + 64) = √100 = 10.
Answer: the hypotenuse is 10.
Task example #2
Condition: calculate the diagonal drawn in a rectangle with a smaller 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 corners. This means that you will need to use one of the formulas in which there are trigonometric functions. First, you need to determine the value of one of sharp corners.
Let the smaller of the angles referred to in the condition be denoted by α. Then the right angle, which 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, the formula described in method No. 3 will be required.
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.
In life, we often have to deal with math problems: at school, at university, and then helping your child with homework. People of certain professions will encounter 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 triangle.
What is a right triangle
First, let's remember what a right triangle is. The right triangle is 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 that form a right angle are called the 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 leg 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 decide: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).
Trigonometric relations to find the leg of a right triangle
It is also possible to find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding the leg using trigonometric functions: by sine, cosine, tangent, cotangent. To solve the problems, the table below will help us. Let's consider these options.
Find the leg of a right triangle using the sine
The sine of an angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin \u003d 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 and angle A is 30 degrees. According to 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 \u003d 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 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 the tangent
The tangent of an angle (tg) is the ratio of the opposite leg to the adjacent one. Formula: tg \u003d a / b, where a is the leg opposite to the corner, and b is adjacent. Let's transform the formula and get: a=tg*b.
Example. Angle A is 45 degrees, the hypotenuse is 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 an angle (ctg) is the ratio of the adjacent leg to the opposite leg. Formula: ctg \u003d b / a, where b is the leg adjacent to the corner, and is opposite. In other words, the cotangent is the "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 the leg in a right triangle. As you can see, it is not so difficult, the main thing is to remember the formulas.
Among the numerous calculations made to calculate certain quantities of various is finding the hypotenuse of the triangle. Recall that a triangle is a polyhedron with three angles. Below are several ways to calculate the hypotenuse of various triangles.
First, let's see how to find the hypotenuse of a right triangle. For those who have forgotten, a right triangle is a triangle with an angle of 90 degrees. The side of the triangle that is on opposite side 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 is 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 foregoing, it follows that when calculating the length of the hypotenuse, it is necessary to square each of the leg values in turn. Then add up the received numbers and extract from the result Square root.
Consider an example: Given a triangle with a right angle. One leg is 3 cm, the other 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.
- Known leg (BK) and the angle adjacent to it, which is formed by the hypotenuse and this leg. How to find the hypotenuse of a triangle? Let us denote the known angle as α. According to the property which says 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 as follows: FB= BK*cos(α).
- The leg (KF) and the same angle α are known, only now it will already 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 angle F equal 30 degrees, the second angle B corresponds to 60 degrees. The leg BK is also known, the length of which corresponds to 8 cm. You can calculate the desired value as follows:
FB=BK/cos60=8 cm.
FB = BK / sin30 = 8 cm.
- Known for (R), circumscribed about a triangle with a right angle. How to find the hypotenuse when considering such a problem? From the properties of a circle circumscribed around a triangle with a right angle, it is known that the center of such a circle coincides with the hypotenuse point dividing it in half. In simple words- the radius corresponds to half of the hypotenuse. Hence the hypotenuse is equal to two radii. FB=2*R. If, however, a similar problem is given, in which not the radius, but the median is known, then one 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 it is necessary 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 legs are the same sides. 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’s difficult to remember all the properties, learn ready-made formulas, substituting known values into which you can calculate the required length of the hypotenuse.
Before you find the hypotenuse of a triangle, you need to figure out 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 of the hypotenuse.
- If the leg is equal to ½ of the value 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. by the most simple solution is the calculation through the legs. Let's say you know the values of the legs of sides A and B. Then the Pythagorean theorem comes to the rescue, telling us that if we square each leg value and sum the data obtained, we will find out what the hypotenuse is. Thus, we just need to extract the square root value:
For example, if leg A = 3 cm and leg B = 4 cm, then the calculation would look like this:
How to find the hypotenuse through an angle?
Another way to help find out what the hypotenuse in a right triangle is equal to is to calculate through a given angle. To do this, we need to derive the value through the sine formula. Suppose we know the value of the leg (A) and the value of the opposite angle (α). Then the whole solution is in one formula: С=А/sin(α).
For example, if the length of the leg is 40 cm and the angle is 45°, then the length of the hypotenuse can be derived as follows:
You can also determine the desired value through the cosine of a given angle. Suppose we know the value of one leg (B) and an acute included angle (α). Then one formula is needed to solve the problem: С=В/ 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 examined the main ways to find out the hypotenuse in a triangle. In the course of solving the task, it is important to focus on the available data, then finding the unknown value will be quite simple. You need to know just a couple of formulas and the process of solving problems will become simple and enjoyable.
The triangle represents 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, according to which the square of the hypotenuse is equal to the sum of the squares of the leg.
If we label the legs with "a" and "b" and the hypotenuse with "c", then pages can be found with the following formulas:
If the acute angles of a right triangle (a and b) are known, its sides can be found with the following formulas:
cropped triangle
A triangle is called an equilateral triangle in which both sides are the same.
How to find the hypotenuse in two legs
If the letter "a" is identical to the same page, "b" is the base, "b" is the corner opposite the base, "a" is the adjacent corner, the following formulas can be used to calculate pages:
Two corners and side
If one page (c) and two angles (a and b) of any triangle are known, the sine formula is used to calculate the remaining pages:
You must find the third value y = 180 - (a + b) because
the sum of all the angles of a triangle is 180°; 
Two sides and an angle
If two sides of a triangle (a and b) and the angle between them (y) are known, the cosine theorem can be used to calculate the third side.
How to determine the perimeter of a right triangle
A triangular triangle is a triangle, one of which is 90 degrees, and the other two are acute. calculation perimeter such triangle depending on the amount of known information about it.
You will need it
- Depending on the occasion, skills 2 of the three sides of the triangle, as well as one of its sharp corners.
instructions
first Method 1. If all three pages are known triangle. Then, whether perpendicular or not triangular, the perimeter is calculated as: P = A + B + C, where possible, c is the hypotenuse; a and b are legs.
second Method 2.
If a rectangle has only two sides, then using the Pythagorean theorem, triangle can be calculated using the formula: P = v (a2 + b2) + a + b or P = v (c2 - b2) + b + c.
third Method 3. Let the hypotenuse be c and an acute angle? Given a right triangle, it will be possible to find the perimeter in this way: P = (1 + sin?
fourth Method 4. They say that in the right triangle the length of one leg is equal to a and, on the contrary, has an acute angle. Then calculate perimeter This triangle will be performed according to the formula: P = a * (1 / tg?
1 / son? + 1)
fifth Method 5.
Triangle Online Calculation
Let our leg lead and be included in it, then the range will be calculated as: P = A * (1 / CTG + 1 / + 1 cos?)
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The Pythagorean theorem is the basis of any mathematics. Specifies the relationship between the sides of a true triangle. Now there are 367 proofs of this theorem.
instructions
first The classic school formulation of the Pythagorean theorem sounds like this: the square of the hypotenuse is equal to the sum of the squares of the legs.
To find the hypotenuse in a right triangle of two Catets, you must turn to square the length of the legs, assemble them, and take the square root of the sum. In the original formulation of his statement, the market is based on the hypotenuse, equal to the sum of the squares of 2 squares produced by Catete. However, the modern algebraic formulation does not require the introduction of a domain representation.
second For example, a right triangle whose legs are 7 cm and 8 cm.
Then, according to the Pythagorean theorem, the square hypotenuse is R + S = 49 + 64 = 113 cm. The hypotenuse is equal to the square root of 113.
Angles of a right triangle
The result was an unreasonable number.
third If the triangles are legs 3 and 4, then the hypotenuse = 25 = 5. When you take the square root, you get natural number. The numbers 3, 4, 5 form a Pygagorean triple, since they satisfy the relation x? +Y? = Z, which is natural.
Other examples of a Pythagorean triplet are: 6, 8, 10; 5, 12, 13; 15, 20, 25; 9, 40, 41.
fourth In this case, if the legs are identical to each other, the Pythagorean theorem turns into a more primitive equation. For example, let such a hand be equal to the number A and the hypotenuse is defined for C, and then c? = Ap + Ap, C = 2A2, C = A? 2. In this case, you don't need A.
fifth Pythagorean theorem - special case, which is greater than the general cosine theorem, which establishes a relationship between the three sides of a triangle for any angle between two of them.
Tip 2: How to determine the hypotenuse for legs and angles
The hypotenuse is called the side in a right triangle that is opposite the 90 degree angle.
instructions
first In the case of well-known catheters, as well as an acute angle of a right triangle, the hypotenuse can have a size equal to the ratio of the leg to the cosine / sine of this angle, if the angle was opposite / e include: H \u003d C1 (or C2) / sin, H \u003d C1 (or С2 ?) / cos ?. Example: Let ABC be given an irregular triangle with hypotenuse AB and right angle C.
Let B be 60 degrees and A 30 degrees. The length of the stem BC is 8 cm. The length of the hypotenuse AB should be found. To do this, you can use one of the above methods: AB = BC / cos60 = 8 cm. AB = BC / sin30 = 8 cm.
The hypotenuse is the longest side of the rectangle triangle. It is located at a right angle. Method for finding the hypotenuse of a rectangle triangle depending on the source data.
instructions
first If your legs are perpendicular triangle, then the length of the hypotenuse of the rectangle triangle can be found by the Pythagorean analogue - the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs: c2 = a2 + b2, where a and b are the length of the legs of the right triangle .
second If it is known and one of the legs is at an acute angle, the formula for finding the hypotenuse will depend on the presence or absence at a certain angle with respect to the known leg - adjacent (the leg is located near), or vice versa (the opposite case is located nego.V of the specified angle is equal to the fraction leg hypotenuse in cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio of sinusoidal angles: da = a / sin.
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Helpful Hints
An angular triangle whose sides are connected as 3:4:5, called the Egyptian delta, due to the fact that these figures were widely used by the architects of ancient Egypt.
This is also the simplest example of Jeron's triangles, with pages and area represented as integers.
A triangle is called a rectangle whose angle is 90°. The side opposite the right corner is called the hypotenuse, the other side is called the legs.
If you want to find how a right triangle is formed by some properties of regular triangles, namely the fact that the sum of the acute angles is 90°, which is used, and the fact that the length of the opposite leg is half the hypotenuse is 30°.
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cropped triangle
One of the properties of an equal triangle is that its two angles are the same.
To calculate the angle of a right equilateral triangle, you need to know that:
- It's no worse than 90°.
- The values of acute angles are determined by the formula: (180 ° -90 °) / 2 = 45 °, i.e.
Angles α and β are 45°.
If the known value of one of the acute angles is known, the other can be found using the formula: β = 180º-90º-α or α = 180º-90º-β.
This ratio is most commonly used if one of the angles is 60° or 30°.
Key Concepts
The sum of the interior angles of a triangle is 180°.
Because it's one level, two stay sharp.
Calculate triangle online
If you want to find them, you need to know that:
other methods
The acute angle values of a right triangle can be calculated from the mean - with a line from a point on the opposite side of the triangle, and the height - the line is a perpendicular drawn from the hypotenuse at a right angle.
Let the median extend from the right corner to the middle of the hypotenuse, and h be the height. In this case it turns out that: 
- sinα = b / (2 * s); sin β = a / (2 * s).
- cosα = a / (2 * s); cos β = b / (2 * s).
- sinα = h / b; sin β = h / a.
Two pages
If the lengths of the hypotenuse and one of the legs are known in a right triangle or from two sides, then trigonometric identities are used to determine the values of acute angles:
- α=arcsin(a/c), β=arcsin(b/c).
- α=arcos(b/c), β=arcos(a/c).
- α = arctan (a / b), β = arctan (b / a).
Length of a right triangle
Area and Area of a Triangle
perimeter
The circumference of any triangle is equal to the sum of the lengths of the three sides. General formula to find triangular triangle:
where P is the circumference of the triangle, a, b and c are its sides.
Perimeter of an equal triangle can be found by successively combining the lengths of its sides, or multiplying the side length by 2 and adding the length of the base to the product.
The general formula for finding an equilibrium triangle will look like this:
where P is the perimeter of an equal triangle, but either b, b are the base.
Perimeter of an equilateral triangle can be found by successively combining the lengths of its sides, or by multiplying the length of any page by 3.
The general formula for finding the rim of equilateral triangles would look like this:
where P is the perimeter of an equilateral triangle, a is any of its sides.
region
If you want to measure the area of a triangle, you can compare it to a parallelogram. Consider triangle ABC:
If we take the same triangle and fix it so that we get a parallelogram, we get a parallelogram with the same height and base as this triangle:
In this case, the common side of the triangles is folded together along the diagonal of the molded parallelogram.
From the properties of a parallelogram. It is known that the diagonals of a parallelogram are always divided into two equal triangles, then the surface of each triangle is equal to half the range of the parallelogram.
Since the area of the parallelogram is the product of its base height, the area of the triangle will be half that product. So for ΔABC the area will be the same
Now consider a right triangle:
Two identical right triangles can be bent into a rectangle if it leans against them, which is every other hypotenuse.
Since the surface of the rectangle coincides with the surface of the adjacent sides, the area of this triangle is the same:
From this we can conclude that the surface of any right triangle is equal to the product of the legs divided by 2.
From these examples, we can conclude that the surface of each triangle is the same as the product of the length, and the height is reduced to the base divided by 2.
The general formula for finding the area of a triangle would look like this:
where S is the area of the triangle, but its base, but the height falls to the bottom a.
