Before you find the hypotenuse of a triangle, you need to figure out 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 of the hypotenuse.
- If the leg is equal to ½ of the value 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. by the most simple solution is the calculation through the legs. Let's say you know the values of the legs of sides A and B. Then the Pythagorean theorem comes to the rescue, telling us that if we square each leg value and sum the data obtained, we will find out what the hypotenuse is. Thus, we just need to extract the square root value:
For example, if leg A = 3 cm and leg B = 4 cm, then the calculation would look like this:
How to find the hypotenuse through an angle?
Another way to help find out what the hypotenuse in a right triangle is equal to is to calculate through a given angle. To do this, we need to derive the value through the sine formula. Suppose we know the value of the leg (A) and the value of the opposite angle (α). Then the whole solution is in one formula: С=А/sin(α).
For example, if the length of the leg 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 of a given angle. Suppose we know the value of one leg (B) and an acute included angle (α). Then one formula is needed to solve the problem: С=В/ 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 examined the main ways to find out the hypotenuse in a triangle. In the course of solving the task, it is important to focus on the available data, then finding the unknown value will be quite simple. You need to know just a couple of formulas and the process of solving problems will become simple and enjoyable.
Knowing one of the legs in a right triangle, you can find the second leg and the hypotenuse using trigonometric relationships - the sine and tangent of a known angle. Since the ratio of the leg opposite the angle to the hypotenuse is equal to the sine of this angle, therefore, in order to find the hypotenuse, the leg must be divided by the sine of the angle. a/c=sinα c=a/sinα
The second leg can be found from the tangent of the known angle, as the ratio of the known leg to the tangent. a/b=tanα b=a/tanα
To calculate not famous angle in a right triangle, you need to subtract the angle α from 90 degrees. β=90°-α
Perimeter and area right triangle through the leg and the angle opposite to it can be expressed by substituting the previously obtained expressions for the second leg and the hypotenuse into the formulas. P=a+b+c=a+a/tanα +a/sinα =a tanα sinα+a sinα+a tanα S=ab/2=a^2/( 2 tanα)
You can also calculate the height through trigonometric relations, but already in the internal right-angled triangle with side a, which it forms. To do this, you need side a, as the hypotenuse of such a triangle, multiplied by the sine of the angle β or the cosine of α, since according to trigonometric identities they are equivalent. (fig. 79.2) h=a cosα
The median of the hypotenuse is equal to half of the hypotenuse or the known leg a divided by two sines α. To find the medians of the legs, we bring the formulas to the appropriate form for the known side and angles. (fig.79.3) m_с=c/2=a/(2 sinα) m_b=√(2a^2+2c^2-b^2)/2=√(2a^2+2a^2+2b^ 2-b^2)/2=√(4a^2+b^2)/2=√(4a^2+a^2/tan^2α)/2=(a√(4 tan^2 α+1))/(2 tanα) m_a=√(2c^2+2b^2-a^2)/2=√(2a^2+2b^2+2b^2-a^2)/ 2=√(4b^2+a^2)/2=√(4b^2+c^2-b^2)/2=√(3 a^2/tan^2α +a^2/sin ^2α)/2=√((3a^2 sin^2α+a^2 tan^2α)/(tan^2α sin^2α))/2=(a√( 3 sin^2α+tan^2α))/(2 tanα sinα)
Since the bisector of a right angle in a triangle is the product of two sides and the root of two, divided by the sum of these sides, replacing one of the legs with the ratio of the known leg to the tangent, we obtain the following expression. Similarly, by substituting the ratio into the second and third formulas, one can calculate the bisectors of the angles α and β. (fig.79.4) l_с=(a a/tanα √2)/(a+a/tanα)=(a^2 √2)/(a tanα+a)=(a√2)/ (tanα+1) l_a=√(bc(a+b+c)(b+c-a))/(b+c)=√(bc((b+c)^2-a^2))/ (b+c)=√(bc(b^2+2bc+c^2-a^2))/(b+c)=√(bc(b^2+2bc+b^2))/(b +c)=√(bc(2b^2+2bc))/(b+c)=(b√(2c(b+c)))/(b+c)=(a/tanα √(2c (a/tanα +c)))/(a/tanα +c)=(a√(2c(a/tanα +c)))/(a+c tanα) l_b=√ (ac(a+b+c)(a+c-b))/(a+c)=(a√(2c(a+c)))/(a+c)=(a√(2c(a+a /sinα)))/(a+a/sinα)=(a sinα √(2c(a+a/sinα)))/(a sinα+a)
The middle line runs parallel to one of the sides of the triangle, while forming another similar right-angled triangle with the same angles, in which all sides are half the size of the original one. Based on this, the middle lines can be found using the following formulas, knowing only the leg and the angle opposite to it. (fig.79.7) M_a=a/2 M_b=b/2=a/(2 tanα) M_c=c/2=a/(2 sinα)
The radius of the inscribed circle is equal to the difference between the legs and the hypotenuse, divided by two, and to find the radius of the circumscribed circle, you need to divide the hypotenuse by two. We replace the second leg and the hypotenuse with the ratios of the leg a to the sine and tangent, respectively. (Fig. 79.5, 79.6) r=(a+b-c)/2=(a+a/tanα -a/sinα)/2=(a tanα sinα+a sinα-a tanα)/(2 tanα sinα) R=c/2=a/2sinα
In life, we often have to deal with math problems: at school, at university, and then helping your child with homework. People of certain professions will encounter 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 triangle.
What is a right triangle
First, let's remember what a right triangle is. The 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. The sides that form a right angle are called the legs, and the side that lies opposite the right angle is called the hypotenuse.
Finding 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 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.” Formula: c²=a²+b², where c is the hypotenuse, a and b are the 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. Next, we decide: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).
Trigonometric relations to find the leg of a right triangle
It is also possible to find an unknown leg if any other side and any sharp corner right triangle. There are four options for finding the leg using trigonometric functions: by sine, cosine, tangent, cotangent. To solve the problems, the table below will help us. Let's consider these options.
Find the leg of a right triangle using the sine
The sine of an angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin \u003d a / c, where a is the leg opposite the given angle, and c is the hypotenuse. Next, we transform the formula and get: a=sin*c.
Example. The hypotenuse is 10 cm and 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∠A*c; a=1/2*10; a=5 (cm).
Find the leg of a right triangle using cosine
The cosine of an angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos \u003d 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 equal to 1/2. Next, we solve: b=cos∠A*c; b=1/2*10, b=5 (cm).
Find the leg of a right triangle using the tangent
The tangent of an angle (tg) is the ratio of the opposite leg to the adjacent one. Formula: tg \u003d a / b, where a is the leg opposite to the corner, and b is adjacent. Let's transform the formula and get: a=tg*b.
Example. Angle A is 45 degrees, the hypotenuse is 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 an angle (ctg) is the ratio of the adjacent leg to the opposite leg. Formula: ctg \u003d b / a, where b is the leg adjacent to the corner, and is opposite. In other words, the cotangent is the "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 the leg in a right triangle. As you can see, it is not so difficult, the main thing is to remember the formulas.
Among the numerous calculations made to calculate certain quantities of various is finding the hypotenuse of the triangle. Recall that a triangle is a polyhedron with three angles. Below are several ways to calculate the hypotenuse of various triangles.
First, let's see how to find the hypotenuse of a right triangle. For those who have forgotten, a right triangle is a triangle with an angle of 90 degrees. The side of the triangle that is on opposite side right angle is called the hypotenuse. In addition, it is the longest side of the triangle. Depending on the known values, the length of the hypotenuse is calculated as follows:
- The lengths of the legs are known. The hypotenuse in this case is calculated using the Pythagorean theorem, which is as follows: the square of the hypotenuse is equal to the sum of the squares of the legs. If we consider a right triangle BKF, where BK and KF are legs, and FB is the hypotenuse, then FB2= BK2+ KF2. From the foregoing, it follows that when calculating the length of the hypotenuse, it is necessary to square each of the leg values in turn. Then add up the received numbers and extract from the result Square root.
Consider an example: Given a triangle with a right angle. One leg is 3 cm, the other 4 cm. Find the hypotenuse. The solution looks like this.
FB2= BK2+ KF2= (3cm)2+(4cm)2= 9cm2+16cm2=25cm2. Extract and get FB=5cm.
- Known leg (BK) and the angle adjacent to it, which is formed by the hypotenuse and this leg. How to find the hypotenuse of a triangle? Let us denote the known angle as α. According to the property which says that the ratio of the length of the leg to the length of the hypotenuse is equal to the cosine of the angle between this leg and the hypotenuse. Considering a triangle, this can be written as follows: FB= BK*cos(α).
- The leg (KF) and the same angle α are known, only now it will already be opposite. How to find the hypotenuse in this case? Let us turn to the same properties of a right triangle and find out that the ratio of the length of the leg to the length of the hypotenuse is equal to the sine of the angle opposite the leg. That is, FB= KF * sin (α).
Let's look at an example. Given the same right triangle BKF with hypotenuse FB. Let angle F equal 30 degrees, the second angle B corresponds to 60 degrees. The leg BK is also known, the length of which corresponds to 8 cm. You can calculate the desired value as follows:
FB=BK/cos60=8 cm.
FB = BK / sin30 = 8 cm.
- Known for (R), circumscribed about a triangle with a right angle. How to find the hypotenuse when considering such a problem? From the properties of a circle circumscribed around a triangle with a right angle, it is known that the center of such a circle coincides with the hypotenuse point dividing it in half. In simple words- the radius corresponds to half of the hypotenuse. Hence the hypotenuse is equal to two radii. FB=2*R. If, however, a similar problem is given, in which not the radius, but the median is known, then one should pay attention to the property of a circle circumscribed around a triangle with a right angle, which says that the radius is equal to the median drawn to the hypotenuse. Using all these properties, the problem is solved in the same way.
If the question is how to find the hypotenuse of an isosceles right triangle, then it is necessary to turn to the same Pythagorean theorem. But, first of all, remember that an isosceles triangle is a triangle that has two identical sides. In the case of a right triangle, the legs are the same sides. We have FB2= BK2+ KF2, but since BK= KF we have the following: FB2=2 BK2, FB= BK√2
As you can see, knowing the Pythagorean theorem and the properties of a right triangle, solving problems in which it is necessary to calculate the length of the hypotenuse is very simple. If it’s difficult to remember all the properties, learn ready-made formulas, substituting known values into which you can calculate the required length of the hypotenuse.
Use a calculator to find the square root of the difference between the squared hypotenuse and the known leg, also squared. The leg is called the side of a right triangle adjacent to the right angle. This expression is derived 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.
Before we look at the various ways to find a leg in a right triangle, let's take some notation. Check which of the listed cases corresponds to the condition of your problem and, depending on this, follow the corresponding paragraph. Find out what quantities in the triangle under consideration are known to you. 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 relationship between the sides and angles of this geometric figure is discussed in detail in mathematical discipline 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 hypotenuse and one of the acute angles adjacent to it are given in the problem, use the Bradys tables.
The inner triangle will be similar to the outer one, since the median lines are parallel to the legs and the hypotenuse, and equal to their halves, respectively. Since the hypotenuse is unknown, to find middle line 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 the right angle. The length of the hypotenuse can be found in various ways. If the length of both legs is known, then its size is calculated by 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 °, we subtract the right angle and the already known one.
When calculating the parameters of a right triangle, it is important to pay attention to known values and solve the problem using the simplest formula. First, let's remember what a right triangle is. A right triangle is a 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 the leg using trigonometric functions: by sine, cosine, tangent, cotangent. The sine of an angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin \u003d 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
The altitude is the perpendicular from any vertex of a triangle to the opposite side (or its extension, for a triangle with an obtuse angle). The heights 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.
Also, it is useful to know the values of trigonometric functions for the most typical angles 30, 45, 60, 90, 180 degrees. If the conditions specify the dimensions of the legs, find the length of the hypotenuse. In life, we often have to face math problems: at school, at university, and then helping our child with homework.
Next, we transform the formula and get: a=sin*c
To solve the problems, the table below will help us. Let's consider these options. interesting special case when one of the acute angles is 30 degrees.
People of certain professions will encounter mathematics on a daily basis.
It is also possible to 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 using various formulas, depending on the number of known variables.
