A right triangle contains a huge number of dependencies. This makes it an attractive object for all kinds of geometric problems. One of the most common problems is finding the hypotenuse.

Right triangle

A right triangle is a triangle that contains a right angle, i.e. 90 degree angle. Only in right triangle can be expressed trigonometric functions through the sizes of the sides. In an arbitrary triangle, additional constructions will have to be made.
In a right triangle, two of the three altitudes coincide with the sides are called legs. The third side is called the hypotenuse. The height drawn to the hypotenuse is the only one in this type of triangle that requires additional construction.

Rice. 1. Types of triangles.

A right triangle cannot have obtuse angles. Just as the existence of a second one is impossible right angle. In this case, the identity of the sum of the angles of a triangle is violated, which is always equal to 180 degrees.

Hypotenuse

Let's move directly to the hypotenuse of the triangle. The hypotenuse is the longest side of a triangle. The hypotenuse is always greater than any of the legs, but it is always less than the sum of the legs. This is a corollary of the triangle inequality theorem.

The theorem states that in a triangle, no side can be greater than the sum of the other two. There is a second formulation or second part of the theorem: in a triangle, opposite the larger side lies the larger angle and vice versa.

Rice. 2. Right triangle.

In a right triangle, the major angle is the right angle, since there cannot be a second right angle or an obtuse angle for the reasons already mentioned. This means that the larger side always lies opposite the right angle.

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

Pythagorean theorem

If a teacher asks about the formula for the hypotenuse of a right triangle, there is a 90% chance that he means the Pythagorean theorem. The theorem states: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

Rice. 3. Hypotenuse of a right triangle.

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

The theorem has the following formula:

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

What have we learned?

We talked about what a right triangle is. We found out why the names of the legs and hypotenuse were invented in the first place. We found out some properties of the hypotenuse and gave the formula for the length of the hypotenuse of a triangle using the Pythagorean theorem.

Test on the topic

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Instructions

The angles opposite to legs a and b will be denoted by A and B, respectively. The hypotenuse, by definition, is the side of a right triangle that is opposite to the right angle (while the hypotenuse forms acute angles with the other sides of the triangle). We denote the length of the hypotenuse by c.

You will need:
Calculator.

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

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

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

Useful tips:
All these expressions are obtained from well-known definitions of trigonometric functions, therefore, even if you forget one of them, you can always quickly derive it using simple operations. It is also useful to know the values ​​of trigonometric functions for the most common angles of 30, 45, 60, 90, 180 degrees.

Before finding the hypotenuse of a triangle, you need to understand what features this figure has. Let's consider the main ones:

1. In a right triangle both acute angles the total will be equal to 90º.
2. A leg lying opposite an angle of 30º will be equal to ½ the size of the hypotenuse.
3. 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.

Knowing one of the legs in a right triangle, you can find the second leg and hypotenuse using trigonometric ratios - sine and tangent of a known angle. Since the ratio of the leg opposite the angle to the hypotenuse is equal to the sine of this angle, therefore, to find the hypotenuse, you need to divide the leg by the sine of the angle. a/c=sin⁡α c=a/sin⁡α

The second leg can be found from the tangent of a known angle, as the ratio of the known leg to the tangent. a/b=tan⁡α b=a/tan⁡α

To calculate the unknown angle in a right triangle, you need to subtract the value of angle α from 90 degrees. β=90°-α

The perimeter and area of ​​a right triangle can be expressed in terms of the leg and the angle opposite it by substituting the previously obtained expressions for the second leg and the hypotenuse into the formulas. P=a+b+c=a+a/tan⁡α +a/sin⁡α =a tan⁡α sin⁡α+a sin⁡α+a tan⁡α S=ab/2=a^2/( 2 tan⁡α)

You can also calculate the height through trigonometric ratios, but in the internal right triangle with side a, which it forms. To do this, you need to multiply side a, as the hypotenuse of such a triangle, by the sine of angle β or cosine α, since according to trigonometric identities they are equivalent. (Fig. 79.2) h=a cos⁡α

The median of the hypotenuse is equal to half the hypotenuse or the known leg a divided by two sines α. To find the medians of the legs, we reduce the formulas to the corresponding form for the known sides and angles. (Fig.79.3) m_с=c/2=a/(2 sin⁡α) m_b=√(2a^2+2c^2-b^2)/2=√(2a^2+2a^2+2b^ 2-b^2)/2=√(4a^2+b^2)/2=√(4a^2+a^2/tan^2⁡α)/2=(a√(4 tan^2⁡ α+1))/(2 tan⁡α) m_a=√(2c^2+2b^2-a^2)/2=√(2a^2+2b^2+2b^2-a^2)/ 2=√(4b^2+a^2)/2=√(4b^2+c^2-b^2)/2=√(3 a^2/tan^2⁡α +a^2/sin ^2⁡α)/2=√((3a^2 sin^2⁡α+a^2 tan^2⁡α)/(tan^2⁡α sin^2⁡α))/2=(a√( 3 sin^2⁡α+tan^2⁡α))/(2 tan⁡α sin⁡α)

Since the bisector of a right angle in a triangle is the product of two sides and the root of two, divided by the sum of these sides, then replacing one of the legs with the ratio of the known leg to the tangent, we obtain the following expression. Similarly, by substituting the ratio into the second and third formulas, you can calculate the bisectors of the angles α and β. (Fig.79.4) l_с=(a a/tan⁡α √2)/(a+a/tan⁡α)=(a^2 √2)/(a tan⁡α+a)=(a√2)/ (tan⁡α+1) l_a=√(bc(a+b+c)(b+c-a))/(b+c)=√(bc((b+c)^2-a^2))/ (b+c)=√(bc(b^2+2bc+c^2-a^2))/(b+c)=√(bc(b^2+2bc+b^2))/(b +c)=√(bc(2b^2+2bc))/(b+c)=(b√(2c(b+c)))/(b+c)=(a/tan⁡α √(2c (a/tan⁡α +c)))/(a/tan⁡α +c)=(a√(2c(a/tan⁡α +c)))/(a+c tan⁡α) l_b=√ (ac(a+b+c)(a+c-b))/(a+c)=(a√(2c(a+c)))/(a+c)=(a√(2c(a+a /sin⁡α)))/(a+a/sin⁡α)=(a sin⁡α √(2c(a+a/sin⁡α)))/(a sin⁡α+a)

The middle line runs parallel to one of the sides of the triangle, while forming another similar right-angled triangle with the same angles, in which all sides are half the size of the original one. Based on this, the middle lines can be found by the following formulas, knowing only the leg and the angle opposite it. (Fig.79.7) M_a=a/2 M_b=b/2=a/(2 tan⁡α) M_c=c/2=a/(2 sin⁡α)

The radius of the inscribed circle is equal to the difference between the legs and the hypotenuse divided by two, and to find the radius of the inscribed circle, you need to divide the hypotenuse by two. We replace the second leg and hypotenuse with the ratio of leg a to sine and tangent, respectively. (Fig. 79.5, 79.6) r=(a+b-c)/2=(a+a/tan⁡α -a/sin⁡α)/2=(a tan⁡α sin⁡α+a sin⁡α-a tan⁡α)/(2 tan⁡α sin⁡α) R=c/2=a/2sin⁡α

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 equal to the ratio 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 in a problem we're talking about o pr O opposite angle, then you need to use it with And well, if - oh pr And lying down, then to O sinus. 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.