The two sides of a right triangle that form a right angle are called legs. Opposite right angle The longest side of the triangle is called the hypotenuse. In order to detect the hypotenuse, you need to know the length of the legs.
Instructions
1. The lengths of the legs and hypotenuse are related by a relationship that is described by the Pythagorean theorem. Algebraic formulation: “In right triangle square of the length of the hypotenuse equal to the sum squares of the lengths of the legs.” The Pythagorean formula looks like this: c2 = a2 + b2, where c is the length of the hypotenuse, a and b are the lengths of the legs.
2. Knowing the lengths of the legs, according to the Pythagorean theorem, it is possible to find the hypotenuse of a right triangle: c = ?(a2 + b2).
3. Example. The length of one of the legs is 3 cm, the length of the other is 4 cm. The sum of their squares is 25 cm?: 9 cm? + 16 cm? = 25 cm?.The length of the hypotenuse in our case is equal to the square root of 25 cm? – 5 cm. Therefore, the length of the hypotenuse is 5 cm.
The hypotenuse is the side in a right triangle that is opposite the 90 degree angle. In order to calculate its length, it is enough to know the length of one of the legs and the size of one of the acute angles of the triangle.
Instructions
1. With the famous leg and acute angle of a right triangle, the size of the hypotenuse can be equal to the ratio leg to the cosine/sine of this angle, if given angle is opposite/adjacent to it: h = C1 (or C2)/sin?; h = C1 (or C2)/cos?. Example: Let a rectangular triangle ABC with hypotenuse AB and right angle C. Let angle B be 60 degrees and angle A 30 degrees. The length of leg BC is 8 cm. We need to find the length of the hypotenuse AB. To do this, you can use any of the methods proposed above: AB = BC/cos60 = 8 cm. AB = BC/sin30 = 8 cm.
The hypotenuse is the longest side of a rectangular triangle. It is located opposite the right angle. Method for finding the hypotenuse of a rectangular triangle depends on what initial data you have.
Instructions
1. If we have rectangular legs triangle, then the length of the hypotenuse of the rectangular triangle can be discovered with the help of the Pythagorean theorem - 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 lengths of the legs of a rectangular triangle .
2. If we draw one of the legs and an acute angle, then the formula for finding the hypotenuse will depend on which angle in relation to the driven leg - adjacent (located near the leg) or opposite (located opposite it. In the case of an adjacent angle, the hypotenuse is equal to the ratio of the leg by the cosine of this angle: c = a/cos?; E is the opposite angle, the hypotenuse is equal to the ratio of the leg to the sine of the angle: c = a/sin?.
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The hypotenuse is the side of a right triangle that lies opposite the right angle. She happens to be largest side right triangle. It can be calculated using the Pythagorean theorem or using the formulas of trigonometric functions.
Instructions
1. The sides of a right triangle that are adjacent to a right angle are called legs. In the figure, the legs are designated AB and BC. Let the lengths of both legs be given. Let us denote them as |AB| and |BC|. In order to find the length of the hypotenuse |AC|, we use the Pythagorean theorem. According to this theorem, the sum of the squares of the legs is equal to the square of the hypotenuse, i.e. in the notation of our figure |AB|^2 + |BC|^2 = |AC|^2. From the formula we find that the length of the hypotenuse AC is found as |AC| = ?(|AB|^2 + |BC|^2) .
2. Let's look at an example. Let the lengths of the legs |AB| be given. = 13, |BC| = 21. By the Pythagorean theorem we find that |AC|^2 = 13^2 + 21^2 = 169 + 441 = 610. In order to obtain the length of the hypotenuse, we need to extract Square root from the sum of the squares of the legs, i.e. from number 610: |AC| =?610. Using the table of squares of integers, we find out that the number 610 is not a perfect square of any integer. In order to obtain the final value of the length of the hypotenuse, we will try to transfer perfect square from under the root sign. To do this, let's factorize the number 610. 610 = 2 * 5 * 61. Looking at the table of primitive numbers, we see that 61 is a primitive number. Consequently, the subsequent reduction of the number?610 is unrealistic. We get the final result |AC| = ?610. If the square of the hypotenuse was equal to, for example, 675, then?675 = ?(3 * 25 * 9) = 5 * 3 * ?3 = 15 * ?3. If a similar cast is allowed, do reverse check– square the total and compare with the initial value.
3. Let us know one of the legs and the angle adjacent to it. To be specific, let these be the side |AB| and angle?. Then we can use the formula for the trigonometric function cosine - the cosine of an angle is equal to the ratio of the adjacent leg to the hypotenuse. Those. in our notation cos? = |AB| / |AC|. From there we get the length of the hypotenuse |AC| = |AB| / cos ?.If we are familiar with the side |BC| and angle?, then we will use the formula to calculate the sine of an angle - the sine of an angle is equal to the ratio opposite leg to the hypotenuse: sin ? = |BC| / |AC|. We find that the length of the hypotenuse is |AC| = |BC| /cos?.
4. For clarity, let's look at an example. Let the length of the leg |AB| be given. = 15. And the angle? = 60°. We get |AC| = 15 / cos 60° = 15 / 0.5 = 30. Let's look at how you can check your result using the Pythagorean theorem. To do this, we need to calculate the length of the second leg |BC|. Using the formula for the tangent of the angle tg? = |BC| / |AC|, we get |BC| = |AB| *tg? = 15 * tan 60° = 15 * ?3. Next we apply the Pythagorean theorem, we get 15^2 + (15 * ?3)^2 = 30^2 => 225 + 675 = 900. The check is completed.
Helpful advice
After calculating the hypotenuse, check whether the resulting value satisfies the Pythagorean theorem.
Translated from Greek language, hypotenuse means “tight”. For correct understanding imagine a bow string connecting the two ends of a flexible stick. Likewise, in a right triangle, the longest side is the hypotenuse, which lies opposite the right angle. It acts as a connector to the other two sides, called legs. To find out how long this “string” is, you need to have the lengths of the legs, or the size of two acute angles. By combining these data, you can calculate the desired value using formulas.
How to find the hypotenuse by legs
The easiest way to calculate is if you know the size of two legs (let's denote one as A, the other as B). Pythagoras himself comes to the rescue and his worldwide famous theorem. She tells us that if we square the length of the legs and add up the calculated values, then as a result we will know the squared value of the length of the hypotenuse. From the above, we conclude: to find the value of the hypotenuse, it is necessary to extract the square root of the total sum of the squares of the legs C = √ (A² + B²). Example: side A=10 cm, side B=20 cm. The hypotenuse is equal to 22.36 cm. The calculation is as follows: √(10²+20²)=√(100+400)= √500≈22.36.
How to find the hypotenuse through an angle
It's a little more difficult to calculate the length of the hypotenuse through specified angle. If you know the size of one of the two legs (denoted by A) and the size of the angle (denoted by α) that lies opposite it, then the size of the hypotenuse is found using trigonometry, and specifically, the sine. All you need to do is divide the value of the known leg by the sine of the angle. C=A/sin(α). Example: the length of leg A = 30 cm, the angle opposite it is 45°, the hypotenuse will be 42.25 cm. The calculation is as follows: 30/sin(45°) = 30/0.71 = 42.25.
Another way is to find the size of the hypotenuse using the cosine. It is used if you know the size of the leg (denoted by B) and acute angle(denoted by α), which is adjacent to it. All you need to do is divide the value of the leg by the sine of the angle. C=B/cos(α). Example: the length of leg B = 30 cm, the angle opposite it is 45°, the hypotenuse will be 42.25 cm. The calculation is as follows: 30/cos(45°) = 30/0.71 = 42.25.
How to find the hypotenuse of an isosceles right triangle
Any self-respecting schoolchild knows that a triangle is isosceles, provided that two of the three sides are equal to each other. These sides are called lateral, and the one that remains is called the base. If one of the angles is 90°, then you have an isosceles right triangle.
Finding the hypotenuse in such a triangle is simple, because it has several properties that will help. The coals adjacent to the base are equal in value, total amount angle values is 180°. This means that the right angle lies opposite the base, which means the base is the hypotenuse, and the sides are the legs.
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 find out 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 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. 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.
This is all possible ways how 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 that is drawn to larger side, is equal to √52. The other median has a length of √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 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. Mathematical notation it 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 are from a famous song that sounded in feature film The Adventures of Electronics is indeed true to Euclid's geometry. After all, legs are two sides that form an angle, degree measure which is equal to 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.
How to find the hypotenuse using the Pythagorean theorem if both sides are known
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:
- x – first leg;
- y – second leg;
- z – hypotenuse.
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:
- Indicate for yourself where the legs are and where the hypotenuse is.
- Square the first leg.
- Square the second leg.
- Add up the resulting values.
- Extract the root of the number obtained in step 4.
How to find the hypotenuse through the sine if the leg and the acute angle opposite it are known
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:
- a – first leg;
- A – acute angle opposite to the leg;
- c- hypotenuse.
Algorithm for finding the hypotenuse using the sine theorem:
- Indicate for yourself a known leg and the angle opposite to it.
- Divide the leg into the opposite corner.
- Get the hypotenuse.
How to find the hypotenuse through the cosine if the leg and the acute angle adjacent to it are known
The ratio of the known side to the acute adjacent corner 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:
- a – second leg;
- B – acute angle adjacent to the second leg;
- c- hypotenuse.
Algorithm for finding the hypotenuse using the cosine theorem:
- Indicate for yourself a known leg and an adjacent angle.
- Divide the leg by the adjacent angle.
- Get the hypotenuse.
How to find the hypotenuse using the Egyptian triangle
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.
- First three numbers: 3-4-5. The legs here are equal to 3 and 4. Then the hypotenuse will definitely be equal to 5. Check: (9+16=25).
- Second triple of numbers: 5-12-13. Here, too, the legs are equal to 5 and 12. Therefore, the hypotenuse will be equal to 13. Check: (25+144=169).
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 best option is the Pythagorean theorem, but it would also not hurt to remember the triplets of numbers that make up “ Egyptian triangle”, because you can save a lot of time if you come across such values.
Geometry - no simple science. She demands to herself special attention and knowledge of exact formulas. This type of mathematics came to us from Ancient Greece and even after several thousand years it does not lose its relevance. Don't think in vain that this is useless item, filling the heads of students and schoolchildren. In fact, geometry is applicable in many areas of life. Without it, no knowledge of geometry can be built architectural structure, cars are not created, spaceships and airplanes. Complex and not very complex road junctions and ruts - all this requires geometric calculations. Yes, even sometimes you cannot make repairs in your room without knowledge elementary formulas. So don't underestimate the importance of this subject. We study the most common formulas that we have to use in many solutions at school. One of them is finding the hypotenuse in a right triangle. To understand this, read below.
Before we start practicing, let's start with the basics and define what the hypotenuse is in a right triangle.
The hypotenuse is one of the sides in a right triangle that is opposite the 90 degree angle (right angle) and is always the longest.
There are several ways to find the length of the desired hypotenuse in a given right triangle.
In the case when the legs are already known to us, we use the Pythagorean theorem, where we add the sum of the squares of two legs, which will be equal to the square of the hypotenuse.
a and b are legs, c is the hypotenuse.
In our case, for a right triangle, accordingly, the formula will be as follows:
If we substitute known numbers legs a and b, let it be a=3 and b=4, then c=√32+42, then we get c=√25, c=5
When we know the length of only one leg, the formula can be transformed to find the length of the second. It looks like this:
In the case when, according to the conditions of the problem, we know leg A and hypotenuse C, then we can calculate the right angle of the triangle, let's call it α.
To do this we use the formula:
Let the second angle we need to calculate be β. Considering that we know the sum of the angles of a triangle, which is 180°, then: β= 180°-90°-α
In the case when we know the values of the legs, we can use the formula to find the value of the acute angle of the triangle:
Depending on the known generally accepted values, the sides of a rectangle can be found from a variety of different formulas. Here are some of them:
When solving problems involving finding unknowns in a right triangle, it is very important to focus on the values you already know and, based on this, substitute them into the required formula. It will be difficult to remember them right away, so we advise you to make a small handwritten hint and paste it into your notebook.
As you can see, if you delve into all the intricacies of this formula, you can easily figure it out. We recommend trying to solve several problems based on this formula. After you see your result, it will become clear to you whether you understood this topic or not. Try not to memorize, but to delve into the material, it will be much more useful. Memorized material is forgotten after the first test, and you will encounter this formula quite often, so first understand it, and then memorize it. If these recommendations are not given positive effect, that is, the meaning in additional classes this topic. And remember: teaching is light, not teaching is darkness!
