Solution of a right triangle. How to find the sides of a right triangle? Basics of geometry How to find out the hypotenuse knowing the legs online

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:

through the height and the side on which it descends;
according to Heron's formula;
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:

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 will 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. The simplest 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.

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

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 a triangle that is 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 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 numbers and take the square root of the result.

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 terms, the radius corresponds to half 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.

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.

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 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 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. 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.