How to find a leg knowing the hypotenuse and leg. How do I find the sides of a right triangle? Fundamentals of Geometry
After studying the topic about right-angled triangles, students often throw out all the information about them from 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 you need to find it. Or the diameter of the circle coincides with the largest side of the triangle, one of the corners of which is straight. And it is impossible to find it without this knowledge.
There are several options for how to find the hypotenuse of a triangle. The choice of the method depends on the initial data set in the condition of 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 do students forget that this formula is used to find the square of the hypotenuse. This means that to find the side itself, you will need to extract the square root. Therefore, the formula for the hypotenuse, which is usually denoted by the letter "c", will look like this:
c = √ (a 2 + b 2), where the letters "a" and "b" are written both legs of a right-angled triangle.
Method number 2: the leg and the angle that is adjacent to it are known
In order to find out how to find the hypotenuse, you need to remember trigonometric functions. Namely the cosine. For convenience, we will assume that the leg "a" and the adjacent angle α are given.
Now you need to remember that the cosine of the angle of a right-angled triangle is equal to the ratio of the two sides. The numerator will contain the value of the leg, and the denominator will contain the hypotenuse. From this it follows that the latter can be calculated using 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 notation for this angle - β, and leave the side the same "a". In this case, you will need another trigonometric function - sine.
As in the previous example, the sine is equal to the ratio of the leg to the hypotenuse. The formula for this method looks like this:
c = a / sin β.
In order not to get confused in trigonometric functions, you can remember a simple mnemonic rule: if the problem is about pr O lying angle, then you need to use with and nus, if - oh pr and lying, then to O sinus. You should pay attention to the first vowels in the 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-angled triangle. It reads as follows. The center of the circle coincides with the midpoint of the hypotenuse. In other words, the largest side of a right-angled triangle is equal to the diagonal of the circle. That is, the doubled radius. The formula for this task will look like this:
c = 2 * r where r is the known radius.
These are all possible ways of how 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 task number 1
Condition: in a right-angled triangle, medians are drawn to both legs. The length of the one that is drawn to the larger side is √52. The other median is √73 long. It is required to calculate the hypotenuse.
Since the medians are drawn in the triangle, they divide the legs into two equal segments. For the convenience of reasoning and the search for how to find the hypotenuse, you need to introduce several designations. Let both halves of the larger leg be marked with the letter "x", and the other - "y".
Now we need to consider two right-angled triangles, the hypotenuses of which are the 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 by them its hypotenuse.
First, you need to raise everything to the second degree. It turns out:
4y 2 + x 2 = 52
2 + 4x 2 = 73.
From the second equation it can be seen that 2 = 73 - 4x 2. This expression must be substituted into the first one and calculate "x":
4 (73 - 4x 2) + x 2 = 52.
After conversion:
292 - 16 x 2 + x 2 = 52 or 15x 2 = 240.
From the last expression x = √16 = 4.
Now you can calculate "y":
y 2 = 73 - 4 (4) 2 = 73 - 64 = 9.
According to the condition data, 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.
Example task number 2
Condition: calculate the diagonal drawn in the rectangle with the smaller side equal to 41. If it is known that it divides the angle by those that relate like 2 to 1.
In this problem, the diagonal of a rectangle is the largest side in a 90 ° triangle. So it all comes down to how to find the hypotenuse.
The problem deals with corners. This means that you will need to use one of the formulas in which trigonometric functions are present. And first you need to determine the size 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:
It is easy to determine α from this equation. It will be equal to 30º. Moreover, it will lie opposite the smaller side of the rectangle. Therefore, you need the formula described in method # 3.
The hypotenuse is equal to the ratio of the leg to the sine of the opposite angle, that is:
41 / sin 30º = 41 / (0.5) = 82.
Answer: The hypotenuse is 82.
The first are the segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is opposite the 90 ° angle. A Pythagorean triangle is one whose sides are equal to natural numbers; their lengths in this case are called "Pythagorean triplets".
Egyptian triangle
In order for the current generation to learn geometry in the form in which it is taught at school now, it has developed for several centuries. The fundamental point is considered the Pythagorean theorem. The sides of the rectangular are known all over the 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) = 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". The interesting thing is that which is inscribed in the figure is equal to one. The name originated around the 5th century BC, when Greek philosophers traveled to Egypt.
When building the pyramids, architects and surveyors used a ratio of 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 shapes
- 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 equality of figures. Taking the sum of the angles into account, it is easy to prove that the second acute angles are also equal. Thus, the triangles are the same in the second characteristic.
- When two figures are superimposed on each other, we rotate them in such a way that, when combined, they become one isosceles triangle. By its property, the sides, or rather, the hypotenuses, are equal, as are the angles at the base, which means that these figures are the same.
On the first basis, 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 in sign II, the essence of which is the equality of the leg and the acute angle.
Right Angle Triangle Properties
The height dropped from the right angle splits 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 by the hypotenuse, is equal to its half. 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-angled triangle, the properties of angles of 30 °, 45 ° and 60 ° apply.
- At an angle of 30 °, it should be remembered that the opposite leg will be equal to 1/2 of the largest side.
- If the angle is 45 °, then the second acute angle is also 45 °. This suggests that the triangle is isosceles, and its legs are the same.
- The property of a 60 ° angle is that the third angle has a degree measure of 30 °.
The area can be easily recognized by one of three formulas:
- through the height and side to which it descends;
- according to Heron's formula;
- on the sides and the corner between them.
The sides of a right-angled triangle, or rather the legs, converge at two heights. In order to find the third, it is necessary to consider the resulting triangle, and then, by the Pythagorean theorem, calculate the required length. In addition to this formula, there is also the ratio of the doubled area and the length of the hypotenuse. The most common expression among students is the former, as it requires less calculations.
Theorems applied to a right triangle
The geometry of a right triangle includes the use of theorems such as:
In life, we often have to deal with math problems: at school, at the university, and then helping our child with homework. People in certain professions will be exposed to 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-angled triangle.
What is a right triangle
First, let's remember what a right triangle is. A right-angled triangle is a geometric figure of three line segments that connect points that do not lie on one straight line, and one of the corners of this figure is 90 degrees. The sides forming a right angle are called legs, and the side that lies opposite the right angle is called the hypotenuse.
Find 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 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 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. Then we decide: a² = 5²-3²; a² = 25-9; a² = 16; a = √16; a = 4 (cm).
Trigonometric ratios to find the leg of a right triangle
You can also find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding a leg using trigonometric functions: sine, cosine, tangent, cotangent. To solve problems, the table below will help us. Let's consider these options.
Find the leg of a right triangle using sine
The sine of the angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin = a / c, where a is the leg opposite a given angle, and c is the hypotenuse. Next, we transform the formula and get: a = sin * c.
Example. The hypotenuse is 10 cm, the 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∠А * c; a = 1/2 * 10; a = 5 (cm).
Find the leg of a right triangle using the cosine
The cosine of the angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos = 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 1/2. Then we decide: b = cos∠A * c; b = 1/2 * 10, b = 5 (cm).
Find the leg of a right triangle using the tangent
The tangent of the angle (tg) is the ratio of the opposite leg to the adjacent leg. Formula: tg = a / b, where a is the leg opposite to the corner, and b is adjacent. We transform the formula and get: a = tg * b.
Example. Angle A is equal to 45 degrees, hypotenuse is equal to 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 the angle (ctg) is the ratio of the adjacent leg to the opposite leg. Formula: ctg = b / a, where b is the leg adjacent to the corner, a is the opposite leg. In other words, a cotangent is an “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 a leg in a right triangle. As you can see, this is not so difficult, the main thing is to remember the formulas.
Instructions
The angles opposite to 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 (while the hypotenuse forms acute angles with the other sides of the triangle). The length of the hypotenuse is denoted 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 obtained 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 statement stands for square root extraction. The "^ 2" sign 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 denoted 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 by the formula a = b * tgA in the case when leg b and the angle opposite to the desired leg are given (we agreed to denote this angle as A).
Note:
If in your task the leg is not found in any of the described ways, 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, therefore, even if you forgot some of them, you can always quickly derive it by simple operations. Also, it is useful to know the values of trigonometric functions for the most typical angles of 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 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 differently, 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 mark the legs with the letters "a" and "b" and the hypotenuse with "c", then the pages can be found with the following formulas:
If the acute angles of a right-angled 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 to the base, "a" is the adjacent corner, the following formulas can be used to calculate the 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 should find the third value y = 180 - (a + b) because
the sum of all the angles of the triangle is 180 °; 
Two sides and an angle
If you know the two sides of the triangle (a and b) and the angle between them (y), 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 sharp. payment perimeter such triangle depending on the amount of known information about it.
You need it
- Depending on the case, skills are 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, regardless, perpendicular or non-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 the rectangle has only two sides, then using the Pythagorean theorem, triangle can be calculated by the formula: P = v (a2 + b2) + a + b or P = v (c2 - b2) + b + c.
third Method 3. Let the hypotenuse c and an acute angle? Given a right-angled triangle, it will be possible to detect the perimeter in this way: P = (1 + sin?
fourth Method 4. It is said 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 is triangle will be performed according to the formula: P = a * (1 / tg?
1 / son? + 1)
fifth Method 5.
Online triangle 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 foundation of any mathematics. Defines the relationship between the sides of a true triangle. Now 367 proofs of this theorem are indicated.
instructions
first The classical 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, collect them, and take the square root of the sum. In the original formulation of his statement, the market is based on a hypotenuse equal to the sum 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-angled triangle whose legs are 7 cm and 8 cm.
Then, according to the Pythagorean theorem, the square hypotenuse is equal to R + S = 49 + 64 = 113 cm.The hypotenuse is equal to the square root of the number 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 Pyghagorean triplet, since they satisfy the relation x? + Y? = Z, which is natural.
Other examples of the 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, suppose such a hand is 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 do not need A.
fifth The Pythagorean theorem is a special case that is larger than the general cosine theorem, which establishes a connection 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 to the 90 degree angle.
instructions
first In the case of known catheters, as well as an acute angle of a right-angled triangle, the hypotenuse size may equal 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 C2?) / Cos?. Example: Let ABC be an irregular triangle with hypotenuse AB and right angle C.
Let B be 60 degrees and A 30 degrees. BC stem length 8 cm. AB hypotenuse length should be found. To do this, you can use one of the above methods: AB = BC / cos60 = 8 cm. AB = BC / sin30 = 8 cm.
Hypotenuse is the longest side of the rectangle triangle... It is located at right angles. Rectangle hypotenuse search method 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 of nego is located. the hypotenuse of the leg at the cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio of the sinusoidal angles: da = a / sin.
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Useful Tips
An angular triangle, the sides of which 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 with an angle of 90 °. The side opposite to the right corner is called the hypotenuse, the other side is called the legs.
If you want to find how a right-angled triangle is formed by some of the 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 corners are the same.
To calculate the angle of a right-angled equal triangle, you need to know that:
- This is no worse than 90 °.
- Acute angle values are determined by the formula: (180 ° -90 °) / 2 = 45 °, i.e.
The angles α and β are equal to 45 °.
If the known value of one of the acute angles is known, the other can be found by the formula: β = 180º-90º-α or α = 180º-90º-β.
This ratio is most often used when one of the angles is 60 ° or 30 °.
Key concepts
The sum of the interior angles of a triangle is 180 °.
Because this is one level, two remain sharp.
Calculate triangle online
If you want to find them, you need to know that:
other methods
The acute angle values of a right-angled 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 dropped from the hypotenuse at a right angle.
Let the median be extended from the right corner to the middle of the hypotenuse, and h is the height. In this case, it turns out that: 
- sin α = b / (2 * s); sin β = a / (2 * s).
- cos α = a / (2 * s); cos β = b / (2 * s).
- sin α = h / b; sin β = h / a.
Two pages
If the lengths of the hypotenuse and one of the legs are known in a right-angled triangle or on both 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 from its side.
Perimeter of an Equal Triangle can be found by concatenating the side lengths sequentially, or by multiplying the side length by 2 and adding the base length 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 is the base.
Perimeter of an equilateral triangle can be found by concatenating the length of its sides sequentially, or by multiplying the length of any page by 3.
The general formula for finding the rim of equilateral triangles will 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 the 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 of the parallelogram range.
Since the area of the parallelogram is the same as the product of its base height, the area of the triangle will be half that product. Thus, for ΔABC, the region will be the same
Now consider a right-angled triangle:
Two identical right-angled triangles can be bent into a rectangle if it leans against them, which is each 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-angled triangle is equal to the product of the legs, divided by 2.
From these examples, it can be inferred that the surface of each triangle is the same as the product of the length, and the height is reduced to a substrate 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.
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