Geometry is not a simple science. It may be useful for both school curriculum, and in real life. Knowledge of many formulas and theorems will simplify geometric calculations. One of the most simple figures in geometry it is a triangle. One of the varieties of triangles, equilateral, has its own characteristics.
By definition, a triangle is a polyhedron that has three angles and three sides. This is a flat two-dimensional figure, its properties are studied in high school. Based on the type of angle, there are acute, obtuse and right triangles. A right triangle is a geometric figure where one of the angles is 90º. Such a triangle has two legs (they create a right angle), and one hypotenuse (it is opposite right angle). Depending on what quantities are known, there are three simple ways Calculate the hypotenuse of a right triangle.
Pythagorean theorem - the oldest way Calculate any side of a right triangle. It sounds like this: “In a right triangle, the square of the hypotenuse equal to the sum squares of legs.” Thus, to calculate the hypotenuse, one should derive Square root from the sum of two legs squared. For clarity, formulas and a diagram are given.
One of the properties of a right triangle states that the ratio of the length of the leg to the length of the hypotenuse is equivalent to the cosine of the angle between this leg and the hypotenuse. Let's call the angle known to us α. Now, thanks to the well-known definition, you can easily formulate a formula for calculating the hypotenuse: Hypotenuse = leg/cos(α)
If the opposite angle is known, it is possible to again use the properties of a right triangle. The ratio of the length of the leg and the hypotenuse is equivalent to the sine of the opposite angle. Let us again call the known angle α. Now for the calculations we will use a slightly different formula:
Hypotenuse = leg/sin (α)
For a deeper understanding of each of the formulas, you should consider illustrative examples. So, suppose given right triangle, where there is such data:
According to the Pythagorean theorem: Hypotenuse = square root of (36+64) = 10 cm.
According to the size of the leg and adjacent angle: 8/0.8 = 10 cm.
According to the size of the leg and the opposite angle: 8/0.8 = 10 cm.
Once you understand the formula, you can easily calculate the hypotenuse with any data.
Video: Pythagorean Theorem
After studying a topic about right triangles, students often forget all the information about them. Including how to find the hypotenuse, not to mention what it is.
And in vain. Because in the future the diagonal of the rectangle turns out to be this very hypotenuse, and it needs to be found. Or the diameter of a circle coincides with the largest side of a triangle, one of the angles of which is right. And it is impossible to find it without this knowledge.
There are several options for finding the hypotenuse of a triangle. The choice of method depends on the initial data set in the problem of quantities.
This is the most memorable method because it uses the Pythagorean theorem. Only sometimes 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 take the square root. Therefore, the formula for the hypotenuse, which is usually denoted by the letter “c,” will look like this:
c = √ (a 2 + b 2), where the letters “a” and “b” represent both legs of a right triangle.
In order to learn how to find the hypotenuse, you will need to remember trigonometric functions. Namely cosine. For convenience, we will assume that leg “a” and the angle α adjacent to it are given.
Now we need to remember that the cosine of the angle of a right triangle is equal to the ratio of the two sides. The numerator will contain the value of the leg, and the denominator will contain the hypotenuse. It follows from this that the latter can be calculated using the formula:
c = a / cos α.
In order not to get confused in the formulas, let’s introduce the designation 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: if in a problem we're talking about o pr O opposite angle, then you need to use it with And well, if - oh pr And lying down, then to O sinus. You should pay attention to the first vowels in keywords. They form pairs o-i or and about.
Now, in order to find out how to find the hypotenuse, you will need to remember the property of the circle that is circumscribed around a right triangle. It reads as follows. The center of the circle coincides with the middle of the hypotenuse. To put it another way, the longest side of a right triangle is equal to the diagonal of the circle. That is, double the radius. The formula for this problem will look like this:
c = 2 * r, where the letter r denotes the known radius.
This is all possible ways how to find the hypotenuse of a right triangle. For each specific task, you need to use the method that is most suitable for the data set.
Condition: in a right triangle, medians are drawn to both sides. The length of the one drawn to the larger side is √52. The other median has length √73. You need to calculate the hypotenuse.
Since medians are drawn in a triangle, they divide the legs into two equal to the segment. For convenience of reasoning and searching for how to find the hypotenuse, you need to introduce several notations. Let both halves of the larger leg be designated by the letter “x”, and the other by “y”.
Now we need to consider two right triangles whose hypotenuses are the known medians. For them you need to write 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 from them its hypotenuse.
First you need to raise everything to the second power. It turns out:
4y 2 + x 2 = 52
y 2 + 4x 2 = 73.
From the second equation it is clear that y 2 = 73 - 4x 2. This expression needs to be substituted into the first one and calculated “x”:
4(73 - 4x 2) + x 2 = 52.
After conversion:
292 - 16 x 2 + x 2 = 52 or 15x 2 = 240.
From last expression x = √16 = 4.
Now you can calculate "y":
y 2 = 73 - 4(4) 2 = 73 - 64 = 9.
According to the conditions, it turns out that the legs of the original triangle are equal to 6 and 8. This means that you can use the formula from the first method and find the hypotenuse:
√(6 2 + 8 2) = √(36 + 64) = √100 = 10.
Answer: hypotenuse equals 10.
Condition: calculate the diagonal drawn in a rectangle with a shorter side equal to 41. If it is known that it divides the angle into those that are related as 2 to 1.
In this problem, the diagonal of a rectangle is the longest side in a 90º triangle. So it all comes down to how to find the hypotenuse.
The problem is about angles. This means that you will need to use one of the formulas that contains trigonometric functions. And first you need to determine the value of one of sharp corners.
Let the smaller of the angles discussed in the condition be designated α. Then the right angle that is divided by the diagonal will be equal to 3α. The mathematical notation for this looks like this:
From this equation it is easy to determine α. It will be equal to 30º. Moreover, it will lie opposite the smaller side of the rectangle. Therefore, you will need the formula described in method No. 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.
“And they tell us that the leg is shorter than the hypotenuse...” These lines from the famous song that was heard in the feature film “The Adventures of Electronics” are indeed correct in Euclid’s geometry. After all, legs are two sides forming an angle whose degree measure is 90 degrees. And the hypotenuse is the longest “stretched” side that connects two legs perpendicular to each other, and lies opposite the right angle. That is why it is possible to find the hypotenuse by legs only in a right triangle, and if the leg were longer than the hypotenuse, then such a triangle would not exist.
The theorem states that the square of the hypotenuse is nothing more than the sum of the squares of the legs: x^2+y^2=z^2, where:
But you just need to find the hypotenuse, and not its square. To do this, extract the root.
Algorithm for finding the hypotenuse using two well-known sides:
The ratio of a known leg to an acute angle lying opposite it is equal to the value of the hypotenuse: a/sin A = c. This is a consequence of the definition of sine:
The ratio of the opposite side to the hypotenuse: sin A = a/c, where:
Algorithm for finding the hypotenuse using the sine theorem:
The ratio of the known leg to the acute adjacent angle is equal to the value of the hypotenuse a/cos B = c. This is a consequence of the definition of cosine: the ratio of the adjacent leg to the hypotenuse: cos B= a/c, where:
Algorithm for finding the hypotenuse using the cosine theorem:
The “Egyptian triangle” is a trio of numbers, knowing which you can save time in finding the hypotenuse or even another unknown leg. The triangle has this name because in Egypt some numbers symbolized the Gods and were the basis for the construction of pyramids and other various structures.
Such numbers help even when they are divided or multiplied by any one number. If the legs are 3 and 4, then the hypotenuse will be equal to 5. If you multiply these numbers by 2, then the hypotenuse will also be multiplied by 2. For example, the triple of numbers 6-8-10 will also fit the Pythagorean theorem and you don’t have to calculate the hypotenuse if you remember these triples of numbers. 
Thus, there are 4 ways to find the hypotenuse using the known legs. The most the best option is the Pythagorean theorem, but it would also not hurt to remember the triplets of numbers that make up the “Egyptian triangle”, because you can save a lot of time if you come across such values.
At the very beginning, let us recall that a triangle is a polyhedron that has 3 angles. How to find the hypotenuse of a right triangle if other dimensions of the triangle are known?
Knowing the properties of a right triangle and the Pythagorean theorem, it is very easy to calculate the length of the hypotenuse. If it is still difficult for you to remember all the properties, then simply learn ready-made formulas into which it is very easy to substitute known values to calculate the length of the hypotenuse.
Instructions
Let one of the legs of a right triangle be known. Suppose |BC| = b. Then we can use the Pythagorean theorem, according to the hypotenuse is equal to the sum of the squares of the legs: a^2 + b^2 = c^2. From this equation we find the unknown side |AB| = a = √ (c^2 - b^2).
Let one of the angles of a right triangle be known, suppose ∟α. Then AB and BC of right triangle ABC can be found using trigonometric functions. So we get: sine ∟α is equal to the ratio of the opposite side sin α = b / c, cosine ∟α is equal to the ratio of the adjacent side to the hypotenuse cos α = a / c. From here we find the required side lengths: |AB| = a = c * cos α, |BC| = b = c * sin α.
Let the ratio of the legs k = a / b be known. We also solve the problem using trigonometric functions. The ratio a / b is nothing more than the cotangent ∟α: the adjacent side ctg α = a / b. In this case, from this equality we express a = b * ctg α. And we substitute a^2 + b^2 = c^2 into the Pythagorean theorem:
b^2 * cotg^2 α + b^2 = c^2. Taking b^2 out of brackets, we get b^2 * (ctg^2 α + 1) = c^2. And from here we easily obtain the length of the leg b = c / √(ctg^2 α + 1) = c / √(k^2 + 1), where k is the given ratio of the legs.
By analogy, if the ratio of the legs b / a is known, we solve the problem using the tangent tan α = b / a. We substitute the value b = a * tan α into the Pythagorean theorem a^2 * tan^2 α + a^2 = c^2. Hence a = c / √(tg^2 α + 1) = c / √(k^2 + 1), where k is the given ratio of the legs.
Let's consider special cases.
∟α = 30°. Then |AB| = a = c * cos α = c * √3 / 2; |BC| = b = c * sin α = c / 2.
∟α = 45°. Then |AB| = |BC| = a = b = c * √2 / 2.
Video on the topic
note
Square roots are extracted with a positive sign, because length cannot be negative. This seems obvious, but this error very common if you solve the problem automatically.
To find the legs of a right triangle, it is convenient to use the reduction formulas: sin β = sin (90° - α) = cos α; cos β = cos (90° - α) = sin α.
Sources:
The relationships between the sides and angles of a right triangle are discussed in the branch of mathematics called trigonometry. To find the sides of a right triangle, it is enough to know the Pythagorean theorem, the definitions of trigonometric functions, and have some means for finding the values of trigonometric functions, for example, a calculator or Bradis tables. Let us consider below the main cases of problems of finding the sides of a right triangle.
You will need
Instructions
If you are given one of the acute angles, for example, A, and the hypotenuse, then the legs can be found from the definitions of the basic trigonometric ones:
a= c*sin(A), b= c*cos(A).
If one of the acute angles, for example, A, and one of the legs, for example, a, is given, then the hypotenuse and the other leg are calculated from the relations: b=a*tg(A), c=a*sin(A).
Helpful advice
If you do not know the value of the sine or cosine of one of the angles necessary for calculation, you can use the Bradis tables; they provide the values of trigonometric functions for a large number of angles. In addition, most modern calculators are capable of calculating sines and cosines of angles.
Sources:
Tre square, one of the angles of which is right (equal to 90°) is called rectangular. Its longest side always lies opposite the right angle and is called the hypotenuse, and the other two sides are called legs. If the lengths of these three sides are known, then find the values of all angles of three square and will not be difficult, since in fact you only need to calculate one of the angles. There are several ways to do this.
Instructions
Use to calculate the quantities (α, β, γ) the definitions of trigonometric functions through a rectangular triangle. Such, for example, for the sine of an acute angle as the ratio of the length of the opposite leg to the length of the hypotenuse. This means that if the lengths of the legs (A and B) and the hypotenuse (C), then, for example, you can find the sine of the angle α lying opposite leg A by dividing the length sides And for the length sides C (hypotenuse): sin(α)=A/C. Having found out the value of the sine of this angle, you can find its value in degrees using the inverse function of the sine - arcsine. That is, α=arcsin(sin(α))=arcsin(A/C). In the same way you can find the size of an acute angle in a triangle. square Yes, but this is not necessary. Since the sum of all angles is three square a is 180°, and in three square If one of the angles is 90°, then the value of the third angle can be calculated as the difference between 90° and the value of the found angle: β=180°-90°-α=90°-α.
Instead of defining the sine, you can use the definition of the cosine of an acute angle, which is formulated as the ratio of the length of the leg adjacent to the desired angle to the length of the hypotenuse: cos(α)=B/C. And here use the reverse trigonometric function(arccosine) to find the angle in degrees: α=arccos(cos(α))=arccos(B/C). After this, as in the previous step, all that remains is to find the value of the missing angle: β=90°-α.
You can use a similar tangent - it is expressed by the ratio of the length of the leg opposite the desired angle to the length of the adjacent leg: tan(α)=A/B. Again, determine the angle in degrees using the inverse trigonometric function -: α=arctg(tg(α))=arctg(A/B). The formula for the missing angle will remain unchanged: β=90°-α.
Video on the topic
A triangle is considered to be right-angled if one of its angles is right. Side triangle located opposite the right angle is called the hypotenuse, and the other two sides- legs. To find the lengths of the sides of a rectangular triangle, you can use several methods.
Instructions
You can find out the third sides, knowing the lengths of the other two sides triangle. This can be done using the Pythagorean theorem, which states that a square of a rectangular triangle the sum of the squares of its legs. (a² = b²+ c²). From here we can express the lengths of all sides of a rectangular triangle:
b² = a² - c²;
c² = a² - b²
For example, for a rectangular triangle the length of the hypotenuse a (18 cm) and one of the legs, for example c (14 cm), is known. To length another side, you need to perform 2 algebraic operations:
c² = 18² - 14² = 324 - 196 = 128 cm
c = √128 cm
Answer: leg length is √128 cm or approximately 11.3 cm
You can resort to if you know the length of the hypotenuse and the size of one of the acute points of a given rectangular triangle. Let the length be c and one of the acute angles be equal to α. In this case, find 2 others sides rectangular triangle it will be possible using the following formulas:
a = с*sinα;
b = с*cosα.
You can give: the length of the hypotenuse is 15 cm, one of the acute angles is 30 degrees. To find the lengths of the other two sides you need to perform 2 steps:
a = 15*sin30 = 15*0.5 = 7.5 cm
b = 15*cos30 = (15*√3)/2 = 13 cm (approx.)
The most non-trivial way to find length sides rectangular triangle- is to express it from the perimeter of a given figure:
P = a + b + c, where P is the perimeter of the rectangular triangle. From this expression it is easy to express length any side of a rectangular triangle.
Knowledge of all three sides directly coal triangle is more than enough to calculate any of its angles. There is so much information that you even have the opportunity to choose which parties to use in the calculations in order to use the trigonometric function that suits you best.
Instructions
If you prefer to deal with the arcsine, use the length of the hypotenuse (C) - the longest sides- and that leg (A) that lies opposite the desired angle (α). Dividing the length of this leg by the length of the hypotenuse will give the value of the sine of the desired angle, and the inverse function of the sine - the arcsine - from the resulting value will restore the value of the angle in . Therefore, use the following in your calculations: α = arcsin(A/C).
To replace arcsine with arccosine, use the length calculations of those sides that form the desired angle (α). One of them will be the hypotenuse (C), and the other will be the leg (B). By definition, the cosine is the length of the leg adjacent to the angle to the length of the hypotenuse, and the angle from the cosine value is the arc cosine function. Use the following calculation formula: α = arccos(B/C).
Can be used in calculations. To do this, you need the lengths of the two short sides - the legs. Tangent of an acute angle (α) in a straight line coal triangle is determined by the ratio of the length of the leg (A) lying opposite it to the length of the adjacent leg (B). By analogy with the options described above, use the following formula: α = arctan(A/B).
Formula
c2=a2+b2, where c is the hypotenuse, a and b are the legs. That is, the hypotenuse will be equal to the square root of the sum of the squares of the legs. To find any of the legs, it is enough to subtract the square of the other leg from the square of the hypotenuse and take the square root from the resulting difference.
Thus, knowing the angle and one of the sides, you can use these formulas to calculate the other side. Both sides are also connected by trigonometric relations. The ratio of the opposite to the adjacent is called tangent, and the ratio of adjacent to the opposite is called cotangent. These relationships can be expressed by the formulas tgA=a/b or ctgA=b/a.
