How to find the sides of a right triangle? Basics of geometry. Solving a right triangle Finding a leg in a right triangle
Using a calculator, extract Square root from the difference of the hypotenuse squared and the known leg, also squared. The leg is the side of a right triangle adjacent to the right angle. This expression comes from the Pythagorean theorem, which states that the square of the hypotenuse of a triangle equal to the sum squares of legs.
Before we look at the various ways to find a leg in right triangle, let's use some notation. Check which of the listed cases corresponds to the condition of your task and, depending on this, follow the appropriate paragraph. Find out what quantities you know in the triangle in question. 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 specifies the hypotenuse and one of the acute angles adjacent to it, use Bradis tables.
The inner triangle will be similar to the outer one, since the middle lines are parallel to the legs and hypotenuse, and are equal to their halves, respectively. Since the hypotenuse is unknown, to find 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 a right angle. The length of the hypotenuse can be found different 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°, subtract the right angle and the already known one.
When calculating the parameters of a right triangle, it is important to pay attention to the known values and solve the problem using the simplest formula. First, let's remember what a right triangle is. A 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 a leg using trigonometric functions: sine, cosine, tangent, cotangent. The sine of an angle (sin) is the ratio of the opposite side to the hypotenuse. Formula: sin=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
Altitude is the perpendicular extending from any vertex of the triangle to the opposite side (or its continuation, for a triangle with an obtuse angle). The altitudes 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.
It is also useful to know the values of trigonometric functions for the most common angles of 30, 45, 60, 90, 180 degrees. If the conditions specify the dimensions of the legs, find the length of the hypotenuse. In life we will often have to deal with math problems: at school, at university, and then helping your child with homework.
Next, we transform the formula and get: a=sin*c
The table below will help us solve problems. Let's consider these options. An interesting special case is when one of the acute angles is equal to 30 degrees.
People in certain professions will encounter mathematics on a daily basis.
You can also 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 by various formulas depending on the number of known variables.
Before finding the hypotenuse of a triangle, you need to understand what features this figure has. Let's consider the main ones:
- In a right triangle, both acute angles add up to 90º.
- A leg lying opposite an angle of 30º will be equal to ½ the size of the hypotenuse.
- If the leg is equal to ½ of the hypotenuse, then the second angle will have the same value - 30º.
There are several ways to find the hypotenuse in a right triangle. The most simple solution is a calculation through legs. Let's say you know the values of the sides A and B. Then the Pythagorean theorem comes to the rescue, telling us that if we square each value of the side and sum up the data obtained, we will find out what the hypotenuse is equal to. So we just need to extract the square root value:
For example, if leg A = 3 cm and leg B = 4 cm, then the calculation will look like this:
How to find the hypotenuse through an angle?
Another way to find out what the hypotenuse is in a right triangle is to calculate through a given angle. To do this, we need to derive the value through the sine formula. Let's say we know the size of the leg (A) and the value of the opposite angle (α). Then the whole solution is contained in one formula: C=A/sin(α).
For example, if the leg length is 40 cm and the angle is 45°, then the length of the hypotenuse can be derived as follows:
You can also determine the desired value through the cosine given angle. Let's say we know the value of one leg (B) and an acute adjacent angle (α). Then to solve the problem you will need one formula: C=B/ cos(α).
For example, if the leg length is 50 cm and the angle is 45°, then the hypotenuse can be calculated as follows:
Thus, we looked at the main ways to find out the hypotenuse in a triangle. When solving a problem, it is important to concentrate on the available data, then finding the unknown quantity will be quite simple. You only need to know a couple of formulas and the process of solving problems will become simple and enjoyable.
A right triangle contains a huge number of dependencies. This makes it an attractive object for all 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. 90 degree angle. Only in a right triangle can one express trigonometric functions through the sizes of the sides. In an arbitrary triangle, additional constructions will have to be made.
In a right triangle, two of the three altitudes 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 construction.
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 is violated, which is always equal to 180 degrees.
Hypotenuse
Let's move directly to the hypotenuse of the triangle. The hypotenuse is the longest side of a 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 corollary of the triangle inequality theorem.
The theorem states that in a triangle, no side can be greater than the sum of the other two. There is a second formulation or second part of the theorem: in a triangle, opposite the larger side lies the larger angle and vice versa.
Rice. 2. Right triangle.
In a right triangle, the major angle is the right angle, since there cannot be a second right angle or an obtuse angle for the reasons already mentioned. This means that the larger side always lies opposite the right angle.
It seems unclear why a right triangle deserves a separate name for each of its sides. In fact, in an isosceles triangle, the sides also have their own names: sides and base. But it is precisely for the legs and hypotenuses that teachers especially like to give 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 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, there is a 90% chance that he means the Pythagorean theorem. The theorem states: 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.
Notice 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 found out why the names of the legs and hypotenuse were invented in the first place. We found out some properties of the hypotenuse and gave the formula for the length of the hypotenuse of a triangle using the Pythagorean theorem.
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A triangle is a 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, which states that the square of the hypotenuse is equal to the sum of the squares of the legs.
If we label the legs as "a" and "b" and the hypotenuse as "c", then the 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 angle opposite the base, "a" is the adjacent angle to calculate pages can use following formulas:
Two corners and a 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 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 information known about it.
You'll need it
- Depending on the case, skills 2 three sides of the triangle, as well as one of its acute angles.
instructions
first Method 1. If all three pages are known triangle Then, whether 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 a rectangle has only two sides, then using the Pythagorean theorem, triangle can be calculated using the formula: P = v (a2 + b2) + a + b or P = v (c2 - b2) + b + c.
third Method 3. Let the hypotenuse be c and an acute angle? Given a right triangle, it will be possible to find the perimeter 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 carried out according to the formula: P = a * (1 / tg?
1/son? + 1)
fifths 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 basis of all mathematics. Determines the relationship between the sides of a true triangle. There are now 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 resort to square the lengths 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 the hypotenuse, which is 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 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 unfounded number.
third If the triangles are legs 3 and 4, then hypotenuse = 25 = 5. When you take the square root, you get natural number. The numbers 3, 4, 5 form a Pygagorean triplet, 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, 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 don't need A.
fifths The Pythagorean theorem is a special case, greater than the general cosine theorem, which establishes the 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 the side in a right triangle that is opposite the 90 degree angle.
instructions
first In the case of known catheters, as well as the acute angle of a right triangle, the hypotenuse can have a size equal to the ratio of the leg to the cosine / sine of this angle, if the angle was opposite / e include: H = C1 (or C2) / sin, H = C1 (or C2?) / 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 a 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 discovered by a 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 one of the legs is known and at an acute angle, the formula for finding the hypotenuse will depend on the presence or absence of under certain angle in relation to the known leg - adjacent (the leg is located close), or vice versa (the opposite case is located nego.V of the specified angle is equal to the fraction of the hypotenuse of the leg in the cosine angle: a = a / cos; E, on the other hand, the hypotenuse is the same as the ratio sinusoidal angles: da = a / sin.
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Useful tips
An angular triangle whose sides are related 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 Jero's triangles, in which pages and area are represented by integers.
A triangle is called a rectangle whose angle is 90°. The side opposite the right corner is called the hypotenuse, the other 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 equal.
To calculate the angle of a right congruent triangle, you need to know that:
- This is 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 equal to 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 often 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 remain sharp.
Calculate triangle online
If you want to find them, you need to know that:
other methods
The values of the acute angles of a right triangle can be calculated from the average - 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 let h be 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 triangle or on both sides, then trigonometric identities are used to determine the values of the 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 of 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 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 sequentially 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 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 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 a parallelogram is the same as the product of its base height, the area of the triangle will be equal to half of this product. Thus, 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 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 triangle is equal to the product of the legs divided by 2.
From these examples it can be concluded that the surface of each triangle is the same as the product of the length, and the height is reduced to the 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.
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.
Method number 1: both sides are given
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.
Method number 2: the leg and the angle adjacent to it are known
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.
Method number 3: given a leg and an angle that lies opposite it
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 the problem deals with pr O opposite angle, then you need to use it with And well, if - oh pr And lying down, then to O sinus. Pay attention to the first vowels in keywords. They form pairs o-i 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 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.
These are all possible ways to find the hypotenuse of a right triangle. Use in every specific task you need the method that is most suitable for the data set.
Example task No. 1
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 segments. 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 the 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.
Example task No. 2
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. First you need to determine the size of one of the acute angles.
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.
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