What is equal to the hypotenus of the equilateral triangle. How to find katets if known hypotenuse

After studying the topic about rectangular triangles, students often emit 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 hypotenuse, and it needs to be found. 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 a triangle hypothen. The choice of method depends on the source dataset in the value of the values \u200b\u200bof the values.

Method number 1: Any category

This is the most memorable method, because it uses Pythagore's theorem. Only sometimes disciples forget that this formula is the square of the hypotenuse. So, to find the side itself, you will need to remove the square root. Therefore, the formula for hypotenuse, which is customary to designate the letter "C" will look like this:

c \u003d √ (and 2 + in 2)where the letters "A" and "B" are recorded by both categories of a rectangular triangle.

Method number 2: Knitting catt and angle, which goes to it

In order to find out how to find the hypotenuse, you will need to recall trigonometric functions. Namely Kosinus. For convenience, we assume that the catat "A" and the angle of α are given to it.

Now we need to remember that the cosine of the angle of the rectangular triangle is equal to the attitude of both sides. The numerator will stand the value of the category, and in the denominator - hypotenuses. It follows from this that the latter can be counted by the formula:

c \u003d A / COS α.

Method at number 3: Dana catat and angle that lies in front of him

In order not to confuse in the formulas, we introduce the designation for this angle - β, and the side will leave the former "A". In this case, another trigonometric function is required - sinus.

As in the previous example, sinus is equal to the ratio of the catech for hypotenuse. The formula of this method looks like this:

c \u003d A / SIN β.

In order not to confuse in trigonometric functions, it is possible to remember the simple mnemonic instigated: if the task is talking about abouttvolezhaya coal, then you need to use with andnus if - about andlying, then to aboutsinus. You should pay attention to the first vowels in keywords. They form a couple o-I. or and about.

Method number 4: By radius of the circle described

Now, in order to learn how to find the hypotenuse, it will be necessary to recall the property of the circle, which is described near the rectangular triangle. It says the following. The center of the circle coincides with the middle of the hypotenuse. If you say differently, the biggest side of the rectangular triangle is equal to the circle diagonal. That is a double radius. The formula for this task will look like this:

c \u003d 2 * Rwhere the letter R is indicated by the famous radius.

These are all possible ways to find a rectangular hypotenus. Each particular task is needed by that method that is more suitable for data set.

Example problem number 1

Condition: Medians have been carried out in a rectangular triangle to both categories. The length of the one that was conducted to the larger side is √52. Another median has a length √73. It is required to calculate the hypotenuse.

Since in the triangle, medians were carried out, they divide catts into two equal segments. For the convenience of reasoning and finding how to find a hypotenuse, you need to enter a few designations. Let both halves of the larger category be denoted by the letter "x", and the other is "y".

Now you need to consider two rectangular triangles, with hypotenuses that are famous medians. For them, you need to record the formula of the Pythagora theorem:

(2y) 2 + x 2 \u003d (√52) 2

(y) 2 + (2x) 2 \u003d (√73) 2.

These two equations form a system with two unknowns. Deciding them, it can be easily found kartets of the initial triangle and its hypotenuse on them.

First you need to build everything in the second degree. It turns out:

4th 2 + x 2 \u003d 52

in 2 + 4x 2 \u003d 73.

From the second equation it can be seen that in 2 \u003d 73 - 4x 2. This expression must be substituted in the first and calculate "X":

4 (73 - 4x 2) + x 2 \u003d 52.

After converting:

292 - 16 x 2 + x 2 \u003d 52 or 15x 2 \u003d 240.

From the last expression x \u003d √16 \u003d 4.

Now you can calculate "U":

in 2 \u003d 73 - 4 (4) 2 \u003d 73 - 64 \u003d 9.

According to the data, it turns out that the ratios of the original triangle are equal to 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: hypotenuse is 10.

Example problem number 2.

Condition: Calculate a diagonal spent in a rectangle with a smaller side of equal to 41. If it is known that it divides the angle to those that relate as 2 to 1.

In this problem, the diagonal of the rectangle is the greatest side in the triangle with an angle of 90º. Therefore, everything comes down to how to find the hypotenuse.

The task is talking about the corners. This means that it will be necessary to use one of the formulas in which trigonometric functions are present. And first it is required to determine the value of one of the sharp corners.

Let the smaller of the corners, which are in question in the condition, will be indicated by α. Then the right angle that is divided by a diagonal will be equal to 3α. The mathematical recording of this looks like this:

From this equation simply define α. It will be equal to 30º. Moreover, it will lie opposite the smaller side of the rectangle. Therefore, the formula described in the method number 3 will be required.

The hypotenuse is equal to the ratio of the catech to the sinus of the opposite angle, that is:

41 / Sin 30º \u003d 41 / (0.5) \u003d 82.

Answer: hypotenuse is 82.

Cates are called two sides of a rectangular triangle that form a straight angle. The opposite direct corner is the longest side of the triangle is called hypotenuse. In order to discover the hypotenuse, you need to know the length of the cathets.

Instruction

1. The lengths of cathets and hypotenuses are associated with the relation, which is described by the Pythagora theorem. Algebraic wording: "In a rectangular triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length of the cathets." The formula of the Pythagora looks like this: C2 \u003d A2 + B2, where C is the length of the hypotenuse, a and b - the length of the cathettes.

2. Knowing the length of cathets, according to the Pythagore theorem, is allowed to detect a rectangular hypothenus: C \u003d? (A2 + B2).

3. Example. The length of one of the cathets is 3 cm, the length of another is 4 cm. The sum of their squares is 25 cm?: 9 cm? + 16 cm? \u003d 25 cm?. The hypotenuse. In our case, is equal to square root from 25 cm? - 5 cm. It became, the length of the hypotenuse is 5 cm.

The hypotenuse is called the side in a rectangular triangle, which is the opposite of an angle of 90 degrees. In order to calculate its length, it is enough to know the length of one of the cathets and the magnitude of one of the sharp corners of the triangle.

Instruction

1. With a famous cathet and acute corner of a rectangular triangle, then the size of the hypotenuse can be equal to the ratio of the cote to the cosine / sinus of this angle, if this angle is the opposite / adjacent: H \u003d C1 (either C2) / SIN ?; H \u003d C1 (or C2 ) /COS?. Example: Let the ABC rectangular triangle with a hypothenoise AB and a direct angle of C and the angle B are 60 degrees, and the angle A 30 degrees of the BC 8 Cate length. To do this, it is allowed to use any of the methods proposed above: AB \u003d BC / COS60 \u003d 8 cm.ab \u003d Bc / sin30 \u003d 8 cm.

Hypotenuse - the longest side of the rectangular triangle . It is located opposite to straight corner. The method of finding a rectangular hypotenuse triangle It depends on what initial data you own.

Instruction

1. If you win a rectangular cattiet triangle , then the length of the hypotenuse of rectangular triangle It can be detected with a subband Pythagoree theorem - the square of the hypotenuse length is equal to the sum of the squares of the spectal lengths: C2 \u003d A2 + B2, where a and b - the length of the rolls of rectangular triangle .

2. If we serve one of the cathettes and a sharp angle, then the formula for finding the hypotenuse will depend on what a given angle with respect to the watchet is adjacent (located near the category) or the opposite (located on the contrary. In the case of the adjacent angle, hypotenuse is equal to the ratio of category On the cosine of this angle: C \u003d A / COS ?; E the angle of the opposite, hypotenuse is equal to the ratio of the category of the corner: C \u003d A / SIN?.

Video on the topic

The hypotenuse is called the side of the rectangular triangle lying on the contrary direct angle. It is the greatest side of the rectangular triangle. It is permitted by the Pythagora theorem or with support for the formulas of trigonometric functions.

Instruction

1. Cates are called the sides of the rectangular triangle, adjacent to the straight corner. In the picture, the cathets are indicated as AB and BC. Let the length of both cathets are specified. Denote them as | AB | and | BC |. In order to detect the length of hypotenuses | AC |, we use the Pythagora theorem. According to this theorem, the sum of the squares of the cathets is equal to the square of the hypotenuse, i.e. In the notation of our drawing | AB | ^ 2 + | BC | ^ 2 \u003d | AC | ^ 2. From the formula we get that the length of the AC hypotenuse is like | AC | \u003d? (| AB | ^ 2 + | BC | ^ 2).

2. Let us see an example. Let the length of the cathets are set | AB | \u003d 13, | BC | \u003d 21. According to Pythagora theorem, we obtain that | AC | ^ 2 \u003d 13 ^ 2 + 21 ^ 2 \u003d 169 + 441 \u003d 610. In order to obtain the length of the hypotenuse, it is necessary to remove the square root from the sum of the squares of the cathets, i.e. from among 610: | AC | \u003d? 610. Using the table of squares of integers, we find out that the number 610 is not a complete square of some integer. In order to obtain the final value of the hypotenuse length, try to transfer a full square from the root sign. To do this, decompose the number 610 for multipliers. 610 \u003d 2 * 5 * 61. In the table of primitive numbers, we see that 61 is the number primitive. Incidentally, the subsequent cause of the number? 610 is unrealistic. We get the final result | AC | \u003d? 610. If the square of the hypotenuse was equal to, for example, 675, then? 675 \u003d? (3 * 25 * 9) \u003d 5 * 3 *? 3 \u003d 15 *? 3 In the event that similar accuracy is permissible, execute the return check - take the result in the square and compare with the initial value.

3. Let us famous for us one of the cathets and angle adjacent to it. For definiteness, let it be cathets | AB | and corner?. Then we can take advantage of the formula for the trigonometric function of the cosine - cosine of the angle is equal to the attitude of the adjacent catech for hypotenuse. Those. In our designations COS? \u003d | AB | / | AC | Panel Get the length of hypotenuse | AC | \u003d | AB | / COS?. If we were famous for us Kartat | BC | and an angle?, then we use the formula for calculating the sine angle - the corner sinus is equal to the attitude of the opposite category to the hypotenuse: SIN? \u003d | BC | / | AC | We get that the length of the hypotenuse is like | AC | \u003d | BC | / COS?

4. For clarity, we will see an example. Let Dana Cate Length | AB | \u003d 15. And the angle? \u003d 60 °. We get | AC | \u003d 15 / Cos 60 ° \u003d 15 / 0.5 \u003d 30. We will see how it is allowed to check your result with the pythagorette theorem. To do this, we need to count the length of the second category | BC |. Using the formula for tangent TG corner? \u003d | BC | / | AC |, Get | BC | \u003d | AB | * TG? \u003d 15 * TG 60 ° \u003d 15 *? 3. Further apply the Pythagore theorem, we obtain 15 ^ 2 + (15 *? 3) ^ 2 \u003d 30 ^ 2 \u003d\u003e 225 + 675 \u003d 900. The test is executed.

Helpful advice
Calculating the hypotenuse, execute the check - whether the obtained value of the Pythagora theorem satisfies.

At the very beginning, we recall that the triangle is a polyhedron who has 3 angle. How to find a rectangular hypotenuzu, if other triangle values \u200b\u200bare known?

Instruction

1. Known length cathets. In this case, the hypotenuse can be calculated using the Pytagora theorem. This theorem sounds like this: the sum of the squares of the cathets is equal to the square of the hypotenuse. From this it follows to calculate the length of hypotenuse, it is necessary to build a square alternately by each category. After that, the figures obtained are folded, and from the general result already remove the square root.
2. How to find a hypotenneuette in a KFB triangle if you know the catat (VC) and adjacent angle to it? The known angle is denoted by α. One of the properties of the rectangular triangle says the following, the ratio of the length of the rectangular triangle ratio to the length of the hypotenuse is equal to the cosine of the angle located between the hypotenurus and this cathet. This can be written as follows: fb \u003d bk * cos (α).
3. Known another catat (kf) and the same corner of α, now it will be opposite. Hypotenuse can also be found if you apply the same properties of a rectangular triangle. Here we get, the ratio of the length of the ratio of the rectangular triangle to the length of its hypotenuse is equal to the sinus of the angle, the opposite cathelet. We write: fb \u003d kf * sin (α).
4. How to find a triangle hypothen, if a circle is described near it, which is known for its radius. From the properties of the circle, which is described around the rectangular triangle it is known that the center has the center with a point of hypotenuse, which shares it in half. In other words, the radius is equal to half of the hypotenuse. This means that two radius make up the hypotenuse: FB \u003d 2 * R.

Knowing the properties of a rectangular triangle and the pepagora theorem, it is very easy to calculate the length of the hypotenuse. If you still find it difficult to remember all properties, then just learn the finished formulas in which it is very easy to substitute known values \u200b\u200bto calculate the length of the hypotenuse.

Geometry is not a simple science. It requires special attention and knowledge of the exact formulas. This kind of mathematics came to us from ancient Greece and even after several thousand years she does not lose their relevance. It is not necessary to think in vain that this is a useless thing that scores the head of students and schoolchildren. In fact, the geometry is applicable in many spheres of life. Without her knowledge of geometry, no architectural structure is not built, cars, cosmic ships and aircraft are created. Complex and not very union of roads and king - it all needs geometric calculations. Yes, even sometimes repairs in your room you can not do without knowing the elementary formulas. So do not underestimate the importance of this subject. The most frequent formulas that have to be used in many decisions, we study at school. One of them is the finding of hypotenuses in a rectangular triangle. To figure it out, read below.

Before proceeding with the practice, let's start with the basics and we define what hypotenuse in a rectangular triangle.

Hypotenuse is one of the sides in a rectangular triangle, which is located opposite an angle of 90 degrees (straight angle) and is always the longest.

There are several ways to find the length of the desired hypotenuse in a given rectangular triangle.

In the case when the catts are already known to us, we use the Pythagore's theorem, where we fold the sum of the squares of two cathets, which will be equal to the square of the hypotenuse.

a and B - Cute, C- hypotenuse.

In our case, for a rectangular triangle, respectively, the formula will be as follows:

If we substitute the known numbers of cathets a and b, let it be a \u003d 3 a b \u003d 4, then C \u003d √32 + 42, then we obtain C \u003d √25, C \u003d 5

When we have the length of only one category, the formula can be converted to find the length of the second. It looks like this:

In the case when, according to the terms of the task, we are known for the catat a and hypotenuse C, then you can calculate the straight angle of the triangle, call it α.

To do this, we use the formula:

Let the second angle that we need to calculate will be β. Given that we know the sum of the corners of the triangle, which is 180 °, then: β \u003d 180 ° -90 ° -α

In the case when we know the values \u200b\u200bof the cathets, you can find the value of the sharp corner of the triangle by the formula:

Depending on the known generally accepted values, the side of the rectangle can be found on a variety of different formulas. Here are some of them:

When solving problems with finding unknown in a rectangular triangle, it is very important to emphasize the attention to you already known to you and, based on this, substitute them in the desired formula. Immediately remember them will be difficult, so we advise you to make a small handwritten prompt and incur in the notebook.

As you can see, if you are in all the subtleties of this formula, you can easily figure it out. We recommend trying to solve several tasks based on this formula. After you see your result, you will become clear, you understood this topic or not. Try not to memorize, but to spit into the material, it will be much more useful. A serrated material is forgotten after the first control, and this formula will be found quite often, so you first understand it, and then memorize. If these recommendations did not give a positive effect, that is, it makes sense in the additional classes of this topic. And remember: learning light, not learning darkness!

There are three options for solving this task. The first - if in the conditions of the problem it is given that the catts are equal (in fact, we have a rectangular anoscele triangle). The second is if some kind of an angle is still given (except an angle of 45%, then we have the same anoscele triangle and return to the first version). And the third - when one of the cathets is known. Consider these options in more detail.

How to find equal cathets, with a well-known hypotenuse

• the first catat (we denote by its letter "A") is equal to the second cathelet ((denote by its letter "B"): a \u003d b;

• size cathets;

In this embodiment, the solution of the problem is based on the use of the Pythagorean theorem. It is applied to rectangular triangles and its main option sounds like: "The square of the hypotenuse is equal to the sum of the squares of the cathets." So, we can be equal to us, we can designate both categories with the same sill: a \u003d b, it means a \u003d a.

1. We substitute our conditional notation in theorem (including the foregoing):
   C ^ 2 \u003d a ^ 2 + a ^ 2,
2. Next, we simplify the formula as much as possible:
   C ^ 2 \u003d 2 * (a ^ 2) - group,
   C \u003d √2 * A - bring both parts of the equation to the square root,
   A \u003d C / √2 - we endure the desired.
3. Substituted this value of the hypotenuse and we obtain a solution:
   a \u003d x / √2

How to find katenets, with known hypotenuse and coal

• hypotenuse (denoted by its letter "C") equal to x cm: C \u003d x;
• angle β is equal to q: β \u003d q;

• size cathets;

To solve this problem, it is necessary to use trigonometric functions. Four more popular two of them:

• the sinus function - the sinus of the desired angle is equal to the attitude of the opposite category to the hypotenuse;
• the cosine function - the cosine of the desired angle is equal to the attitude of the adjacent catech for hypotenuse;

You can use any. I will appear an example using the first. Let the katenets we indicate the characters "a" (adjacent to the corner) and "b" (opposite to the corner). Accordingly, our angle lies between cathet "a" and hypotenuse.

1. We substitute the selected conventions in the formula:
   sinβ \u003d b / c
2. We bring catat:
   b \u003d C * SINβ
3. We substitute our given and we have one catat.
   b \u003d C * SINQ

The second catat can be found using the second trigonometric function, or go to the third option.

How to find one catat if hypotenuse is known and other catat

• hypotenuse (denoted by its letter "C") equal to x cm: C \u003d x;
• catat (we denote by its letter "b") is equal to y cm: b \u003d y;

• the size of another category (we denote by its letter "A");

In this embodiment, the solution of the problem, as in the first, is the use of the Pythagores theorem.

1. We substitute our conditional notation in theorem:
   C ^ 2 \u003d a ^ 2 + b ^ 2,
2. We carry out the necessary catat:
   a ^ 2 \u003d C ^ 2-b ^ 2
3. Believe both parts of the equation to the square root:
   a \u003d √ (C ^ 2-b ^ 2)
4. We substitute these values \u200b\u200band have a solution:
   a \u003d √ (x ^ 2-y ^ 2)