How to calculate the length of the hypotenuse. Right Triangle Solution

The right triangle contains a huge number of dependencies. This makes it an attractive object for various 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. angle of 90 degrees. Only in a right triangle can trigonometric functions be expressed in terms of the sides. In an arbitrary triangle, additional constructions will have to be made.
In a right triangle, two of the three heights 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 constructions.

Rice. 1. Types of triangles.

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

Hypotenuse

Let's go directly to the hypotenuse of the triangle. The hypotenuse is the longest side of the 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 consequence of the triangle inequality theorem.

The theorem says that in a triangle, none of the sides can be greater than the sum of the other two. There is also a second formulation or the second part of the theorem: in a triangle, opposite the larger side, there is a larger angle and vice versa.

Rice. 2. Right triangle.

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

It seems incomprehensible why exactly a right-angled triangle deserved a separate name for each of the sides. In fact, in an isosceles triangle, the sides also have their own names: the sides and the base. But it is for the legs and hypotenuses that teachers especially like to put 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-angled triangles and, along with this knowledge, left a whole layer of information on which modern science is built. 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, then with a probability of 90%, he means the Pythagorean theorem. The theorem says: 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.

Pay attention to 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 learned why they came up with the names of the legs and the hypotenuse. We found out some properties of the hypotenuse and gave the formula for the length of the hypotenuse of a triangle through the Pythagorean theorem.

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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 of a 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 about opposite corner, then you need to use with and nous if - oh pr and lying, then to about 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. And first you need to determine the value of one of the acute angles.

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.

A triangle is a geometric number made up 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.

the 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.

the 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 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 The Pythagorean theorem is a special case that is larger 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 size can be 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 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. The general formula for finding a triangular triangle is:

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.

Knowing one of the legs in a right triangle, you can find the second leg and the hypotenuse using trigonometric relationships - the 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, in order to find the hypotenuse, the leg must be divided by the sine of the angle. a/c=sin⁡α c=a/sin⁡α

The second leg can be found from the tangent of the 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 angle α from 90 degrees. β=90°-α

The perimeter and area of ​​\u200b\u200ba right triangle through the leg and the angle opposite to it can be expressed by substituting the previously obtained expressions for the second leg and 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 relations, but already in the internal right-angled triangle with side a, which it forms. To do this, you need side a, as the hypotenuse of such a triangle, multiplied by the sine of the angle β or the cosine of 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 of the hypotenuse or the known leg a divided by two sines α. To find the medians of the legs, we bring the formulas to the appropriate form for the known side 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, 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, one 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 using the following formulas, knowing only the leg and the angle opposite to 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 circumscribed circle, you need to divide the hypotenuse by two. We replace the second leg and the hypotenuse with the ratios of the leg a to the 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

The first are segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is opposite the 90 degree angle. A Pythagorean triangle is one whose sides are equal to natural numbers; their lengths in this case are called the "Pythagorean triple".

egyptian triangle

In order for the current generation to learn geometry in the form in which it is taught at school now, it has been developed for several centuries. The fundamental point is the Pythagorean theorem. The sides of a rectangle are known to the whole world) are 3, 4, 5.

Few people are not familiar with the phrase "Pythagorean pants are equal in all directions." However, in fact, the theorem sounds like this: c 2 (the square of the hypotenuse) \u003d a 2 + b 2 (the sum of the squares of the legs).

Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called "Egyptian". It is interesting 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 on which 12 knots were tied. In this case, the probability of constructing a right-angled triangle increased to 95%.

Signs of equality of figures

• An acute angle in a right triangle and a large side, which are equal to the same elements in the second triangle, is an indisputable sign of the equality of the 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 in the second criterion.
• When two figures are superimposed on each other, we rotate them in such a way that, when combined, they become one isosceles triangle. According to its property, the sides, or rather, the hypotenuses, are equal, as well as the angles at the base, which means that these figures are the same.

By the first sign, it is very easy to prove that the triangles are really equal, the main thing is that the two smaller sides (i.e., the legs) are equal to each other.

The triangles will be the same according to the II sign, the essence of which is the equality of the leg and the acute angle.

Right angle triangle properties

The height, which was lowered from a right angle, divides the figure into two equal parts.

The sides of a right triangle and its median are easy to recognize by the rule: the median, which is lowered to 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 o, 45 o and 60 o apply.

• At an angle that is 30 °, it should be remembered that the opposite leg will be equal to 1/2 of the largest side.
• If the angle is 45o, then the second acute angle is also 45o. This suggests that the triangle is isosceles, and its legs are the same.
• The property of an angle of 60 degrees is that the third angle has a measure of 30 degrees.

The area is easy to find by one of three formulas:

1. through the height and the side on which it descends;
2. according to Heron's formula;
3. along the sides and the angle between them.

The sides of a right triangle, or rather the legs, converge with two heights. 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 the ratio of twice the area and the length of the hypotenuse. The most common expression among students is the first, as it requires less calculations.

Theorems that apply to a right triangle

The geometry of a right triangle includes the use of theorems such as: