How to find the second leg and hypotenuse. Solution of a right triangle. Trigonometric relations to find the leg of a right triangle
Use a calculator to find the square root of the difference between the squared hypotenuse and the known leg, also squared. The leg is called the side of a right triangle adjacent to the right angle. This expression is derived from the Pythagorean theorem, which states that the square of the hypotenuse of a triangle is equal to the sum of the squares of the legs.
Before we look at the various ways to find a leg in a right triangle, let's take some notation. Check which of the listed cases corresponds to the condition of your problem and, depending on this, follow the corresponding paragraph. Find out what quantities in the triangle under consideration are known to you. Use the following expression to calculate the leg: a=sqrt(c^2-b^2), if you know the values of the hypotenuse and the other leg.
The relationships between the sides and angles of this geometric figure are discussed in detail in the mathematical discipline of trigonometry. To apply this equation, you need to know the length of any two sides of a right triangle.
Calculate the length of one of the legs, if the dimensions of the hypotenuse and the other leg are known. If the hypotenuse and one of the acute angles adjacent to it are given in the problem, use the Bradys tables.
The inner triangle will be similar to the outer one, since the median lines are parallel to the legs and the hypotenuse, and equal to their halves, respectively. Since the hypotenuse is unknown, to find the midline M_c, you need to substitute the radical from the Pythagorean theorem.
The hypotenuse is the longest side of a right triangle. It lies opposite the right angle. The length of the hypotenuse can be found in various ways. If the length of both legs is known, then its size is calculated using the Pythagorean theorem: the sum of the squares of the two legs is equal to the square of the hypotenuse. Knowing that the sum of all angles is 180 °, we subtract the right angle and the already known one.
When calculating the parameters of a right triangle, it is important to pay attention to known values and solve the problem using the simplest formula. First, let's remember what a right triangle is. A right triangle is a 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. There are several ways to find out the length of the leg.
Formula: c²=a²+b², where c is the hypotenuse, a and b are the legs
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." There are four options for finding the leg using trigonometric functions: by sine, cosine, tangent, cotangent. 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.
The unusual properties of right triangles were discovered by the ancient Greek scientist Pythagoras, who discovered that the square of the hypotenuse in such triangles is equal to the sum of the squares of the legs
The altitude is the perpendicular from any vertex of a triangle to the opposite side (or its extension, for a triangle with an obtuse angle). The heights of a triangle intersect at one point, which is called the orthocenter. If it is an arbitrary right triangle, then there is not enough data.
Also, it is useful to know the values of trigonometric functions for the most typical angles 30, 45, 60, 90, 180 degrees. If the conditions specify the dimensions of the legs, find the length of the hypotenuse. In life, we often have to face math problems: at school, at university, and then helping our child with homework.
Next, we transform the formula and get: a=sin*c
To solve the problems, the table below will help us. Let's consider these options. An interesting special case is when one of the acute angles is equal to 30 degrees.
People of certain professions will encounter mathematics on a daily basis.
It is also possible to find an unknown leg if any other side and any acute angle of a right triangle are known. Find the side of a right triangle using the Pythagorean theorem. Also, the sides of a right triangle can be found using various formulas, depending on the number of known variables.
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-angled triangle and a large side, which are equal to the same elements in the second triangle, are 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-angled 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:
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 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.
Instruction
The angles opposite the legs a and b will be denoted by A and B, respectively. The hypotenuse, by definition, is the side of a right-angled triangle that is opposite to the right angle (at the same time, the hypotenuse forms acute angles with other sides of the triangle). Let us denote the length of the hypotenuse by s.
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 is derived from the Pythagorean theorem, which states that the square of the hypotenuse of a triangle is equal to the sum of the squares of the 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 the desired leg (we designated this angle as A).
Use the expression a=c*cosB to find the leg if you know the hypotenuse (c) and the angle adjacent to the desired leg (we designated this angle as B).
Calculate the leg using the formula a = b * tgA in the case when the leg b and the angle opposite the desired leg are given (we agreed to denote this angle A).
Note:
If in your task the leg is not found by any of the described methods, most likely it can be reduced to one of them.
Helpful Hints:
All these expressions are obtained from the well-known definitions of trigonometric functions, so even if you forgot one of them, you can always quickly derive it with simple operations. Also, it is useful to know the values of trigonometric functions for the most typical angles 30, 45, 60, 90, 180 degrees.
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, 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 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 angled 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.