Using a calculator, extract Square root from the difference of the hypotenuse squared 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 problem is given the hypotenuse and one of the adjacent sharp corners, 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 by 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. The right triangle is geometric figure of three segments that connect points that do not lie on the same straight line, and one of the angles of this figure is 90 degrees. 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 deal with mathematical 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.
In life, we often have to face math problems: at school, at university, and then helping our child with homework. People of certain professions will encounter mathematics on a daily basis. Therefore, it is useful to memorize or recall mathematical rules. In this article, we will analyze one of them: finding the leg of a right triangle.
What is a right triangle
First, let's remember what a right triangle is. 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. The sides that form a right angle are called the legs, and the side that lies opposite the right angle is called the hypotenuse.
Finding the leg of a right triangle
There are several ways to find out the length of the leg. I would like to consider them in more detail.
Pythagorean theorem to find the leg of a right triangle
If we know the hypotenuse and the leg, then we can find the length of the unknown leg using the Pythagorean theorem. It sounds like this: “The square of the hypotenuse is equal to the sum of the squares of the legs.” Formula: c²=a²+b², where c is the hypotenuse, a and b are the legs. We transform the formula and get: a²=c²-b².
Example. The hypotenuse is 5 cm, and the leg is 3 cm. We transform the formula: c²=a²+b² → a²=c²-b². Next, we decide: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).
Trigonometric relations to find the leg of a right triangle
It is also possible to find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding the leg using trigonometric functions: by sine, cosine, tangent, cotangent. To solve the problems, the table below will help us. Let's consider these options.
Find the leg of a right triangle using the sine
The sine of an angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin \u003d a / c, where a is the leg opposite the given angle, and c is the hypotenuse. Next, we transform the formula and get: a=sin*c.
Example. The hypotenuse is 10 cm and angle A is 30 degrees. According to the table, we calculate the sine of angle A, it is equal to 1/2. Then, using the transformed formula, we solve: a=sin∠A*c; a=1/2*10; a=5 (cm).
Find the leg of a right triangle using cosine
The cosine of an angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos \u003d b / c, where b is the leg adjacent to the given angle, and c is the hypotenuse. Let's transform the formula and get: b=cos*c.
Example. Angle A is 60 degrees, the hypotenuse is 10 cm. According to the table, we calculate the cosine of angle A, it is equal to 1/2. Next, we solve: b=cos∠A*c; b=1/2*10, b=5 (cm).
Find the leg of a right triangle using the tangent
The tangent of an angle (tg) is the ratio of the opposite leg to the adjacent one. Formula: tg \u003d a / b, where a is the leg opposite to the corner, and b is adjacent. Let's transform the formula and get: a=tg*b.
Example. Angle A is 45 degrees, the hypotenuse is 10 cm. According to the table, we calculate the tangent of angle A, it is equal to Solve: a=tg∠A*b; a=1*10; a=10 (cm).
Find the leg of a right triangle using the cotangent
The cotangent of an angle (ctg) is the ratio of the adjacent leg to the opposite leg. Formula: ctg \u003d b / a, where b is the leg adjacent to the corner, and is opposite. In other words, the cotangent is the "inverted tangent". We get: b=ctg*a.
Example. Angle A is 30 degrees, the opposite leg is 5 cm. According to the table, the tangent of angle A is √3. Calculate: b=ctg∠A*a; b=√3*5; b=5√3 (cm).
So, now you know how to find the leg in a right triangle. As you can see, it is not so difficult, the main thing is to remember the formulas.
A 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 one express trigonometric functions through the dimensions 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 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, 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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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=sinα c=a/sinα
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=tanα b=a/tanα
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/tanα +a/sinα =a tanα sinα+a sinα+a tanα S=ab/2=a^2/( 2 tanα)
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 cosα
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 sinα) 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^2α)/2=(a√(4 tan^2 α+1))/(2 tanα) 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^2α +a^2/sin ^2α)/2=√((3a^2 sin^2α+a^2 tan^2α)/(tan^2α sin^2α))/2=(a√( 3 sin^2α+tan^2α))/(2 tanα sinα)
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/tanα √2)/(a+a/tanα)=(a^2 √2)/(a tanα+a)=(a√2)/ (tanα+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/tanα √(2c (a/tanα +c)))/(a/tanα +c)=(a√(2c(a/tanα +c)))/(a+c tanα) l_b=√ (ac(a+b+c)(a+c-b))/(a+c)=(a√(2c(a+c)))/(a+c)=(a√(2c(a+a /sinα)))/(a+a/sinα)=(a sinα √(2c(a+a/sinα)))/(a sinα+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 tanα) M_c=c/2=a/(2 sinα)
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/tanα -a/sinα)/2=(a tanα sinα+a sinα-a tanα)/(2 tanα sinα) R=c/2=a/2sinα
The triangle represents geometric number, consisting of three segments that connect three points that do not lie on the same line. The points that form a triangle are called its points, and the segments are side by side.
Depending on the type of triangle (rectangular, monochrome, etc.), you can calculate the side of the triangle in different ways, depending on the input data and the conditions of the problem.
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To calculate the sides of a right triangle, the Pythagorean theorem is used, according to which the square of the hypotenuse is equal to the sum of the squares of the leg.
If we label the legs with "a" and "b" and the hypotenuse with "c", then pages can be found with the following formulas:
If the acute angles of a right triangle (a and b) are known, its sides can be found with the following formulas:
cropped triangle
A triangle is called an equilateral triangle in which both sides are the same.
How to find the hypotenuse in two legs
If the letter "a" is identical to the same page, "b" is the base, "b" is the corner opposite the base, "a" is the adjacent corner, the following formulas can be used to calculate pages:
Two corners and side
If one page (c) and two angles (a and b) of any triangle are known, the sine formula is used to calculate the remaining pages:
You must find the third value y = 180 - (a + b) because
the sum of all the angles of a triangle is 180°; 
Two sides and an angle
If two sides of a triangle (a and b) and the angle between them (y) are known, the cosine theorem can be used to calculate the third side.
How to determine the perimeter of a right triangle
A triangular triangle is a triangle, one of which is 90 degrees, and the other two are acute. calculation perimeter such triangle depending on the amount of known information about it.
You will need it
- Depending on the occasion, skills 2 of the three sides of the triangle, as well as one of its sharp corners.
instructions
first Method 1. If all three pages are known triangle. Then, whether perpendicular or not triangular, the perimeter is calculated as: P = A + B + C, where possible, c is the hypotenuse; a and b are legs.
second Method 2.
If a rectangle has only two sides, then using the Pythagorean theorem, triangle can be calculated using the formula: P = v (a2 + b2) + a + b or P = v (c2 - b2) + b + c.
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 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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An angular triangle whose sides are connected as 3:4:5, called the Egyptian delta, due to the fact that these figures were widely used by the architects of ancient Egypt.
This is also the simplest example of Jeron's triangles, with pages and area represented as integers.
A triangle is called a rectangle whose angle is 90°. The side opposite the right corner is called the hypotenuse, the other side is called the legs.
If you want to find how a right triangle is formed by some properties of regular triangles, namely the fact that the sum of the acute angles is 90°, which is used, and the fact that the length of the opposite leg is half the hypotenuse is 30°.
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cropped triangle
One of the properties of an equal triangle is that its two angles are the same.
To calculate the angle of a right equilateral triangle, you need to know that:
- It's no worse than 90°.
- The values of acute angles are determined by the formula: (180 ° -90 °) / 2 = 45 °, i.e.
Angles α and β are 45°.
If the known value of one of the acute angles is known, the other can be found using the formula: β = 180º-90º-α or α = 180º-90º-β.
This ratio is most commonly used if one of the angles is 60° or 30°.
Key Concepts
The sum of the interior angles of a triangle is 180°.
Because it's one level, two stay sharp.
Calculate triangle online
If you want to find them, you need to know that:
other methods
The acute angle values of a right triangle can be calculated from the mean - with a line from a point on the opposite side of the triangle, and the height - the line is a perpendicular drawn from the hypotenuse at a right angle.
Let the median extend from the right corner to the middle of the hypotenuse, and h be the height. In this case it turns out that: 
- sinα = b / (2 * s); sin β = a / (2 * s).
- cosα = a / (2 * s); cos β = b / (2 * s).
- sinα = h / b; sin β = h / a.
Two pages
If the lengths of the hypotenuse and one of the legs are known in a right triangle or from two sides, then trigonometric identities are used to determine the values of acute angles:
- α=arcsin(a/c), β=arcsin(b/c).
- α=arcos(b/c), β=arcos(a/c).
- α = arctan (a / b), β = arctan (b / a).
Length of a right triangle
Area and Area of a Triangle
perimeter
The circumference of any triangle is equal to the sum of the lengths of the three sides. General formula to find triangular triangle:
where P is the circumference of the triangle, a, b and c are its sides.
Perimeter of an equal triangle can be found by successively combining the lengths of its sides, or multiplying the side length by 2 and adding the length of the base to the product.
The general formula for finding an equilibrium triangle will look like this:
where P is the perimeter of an equal triangle, but either b, b are the base.
Perimeter of an equilateral triangle can be found by successively combining the lengths of its sides, or by multiplying the length of any page by 3.
The general formula for finding the rim of equilateral triangles would look like this:
where P is the perimeter of an equilateral triangle, a is any of its sides.
region
If you want to measure the area of a triangle, you can compare it to a parallelogram. Consider triangle ABC:
If we take the same triangle and fix it so that we get a parallelogram, we get a parallelogram with the same height and base as this triangle:
In this case, the common side of the triangles is folded together along the diagonal of the molded parallelogram.
From the properties of a parallelogram. It is known that the diagonals of a parallelogram are always divided into two equal triangles, then the surface of each triangle is equal to half the range of the parallelogram.
Since the area of the parallelogram is the product of its base height, the area of the triangle will be half that product. So for ΔABC the area will be the same
Now consider a right triangle:
Two identical right triangles can be bent into a rectangle if it leans against them, which is every other hypotenuse.
Since the surface of the rectangle coincides with the surface of the adjacent sides, the area of this triangle is the same:
From this we can conclude that the surface of any right triangle is equal to the product of the legs divided by 2.
From these examples, we can conclude that the surface of each triangle is the same as the product of the length, and the height is reduced to the base divided by 2.
The general formula for finding the area of a triangle would look like this:
where S is the area of the triangle, but its base, but the height falls to the bottom a.