# How to find the hypotenuse of a right triangle

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Among the numerous calculations made for the calculation of certain quantities of different geometric shapes, have to find the hypotenuse of the triangle.Recall that a triangle is called a polyhedron having three angles.Below are a few different ways to calculate the hypotenuse of triangles.

initially look at how to find the hypotenuse of a right triangle.For those rusty, called rectangular triangle having an angle of 90 degrees.Side triangle located on the opposite side of the right angle is called the hypotenuse.Besides, it is the longest side of the triangle.Depending on the length of the hypotenuse known quantities is calculated as follows:

• known length of the legs.Hypotenuse in this case is calculated using the Pythagorean theorem, which reads as follows: the square of the hypotenuse equals the sum of the squares of the other two sides.If we consider the right triangle BKF, where BK, and legs of KF and FB - the hypotenuse, the FB2 = BK2 + KF2.It follows that in calculating the length of the hypotenuse should be raised successively in each of the squared values ​​of the other two sides.Then add up the numbers and the teachings of the result of the square root.

Consider this example: Given a triangle with a right angle.One leg is 3 cm, the other 4cm.Find the hypotenuse.The solution is as follows.

FB2 = BK2 + KF2 = (3cm) 2+ (4 cm) + 2 = 9sm2 16sm2 = 25 cm2.Square roots and get FB = 5cm.

• known leg (BK) and the angle adjacent to it, which forms the hypotenuse and that the leg.How to find the hypotenuse of the triangle?Let a known angle α.According to the property of a right triangle, which states that the ratio of the length of the leg to the length of the hypotenuse is equal to the cosine of the angle between the leg and a hypotenuse.Considering this triangle can be written as: FB = BK * cos (α).
• known leg (KF) and the same angle α, only now he is opposing.How to find the hypotenuse in this case?Let us all to the same properties of a right triangle and find that the ratio of the length of the leg to the length of the hypotenuse is equal to the sine of the angle of the opposing side.That is FB = KF * sin (α).

Consider an example.Dan is still the same right-angled triangle with a hypotenuse BKF FB.Let the angle F is equal to 30 degrees, the second angle B corresponds to 60 degrees.More known leg BK, the length of which corresponds to 8 cm. Calculate the required quantity may be because:

FB = BK / cos60 = 8 cm.
FB = BK / sin30 = 8 cm.

• known circle radius (R),described about the triangle with the right angle.How to find the hypotenuse in the consideration of such a problem?From the properties of the circle circumscribed around a triangle with a right angle is known, such that the center of the circle coincides with the point of the hypotenuse dividing it in half.In simple terms - the radius is half the hypotenuse.Hence the hypotenuse is equal to twice the radius.FB = 2 * R.If given a similar problem, which is not known to the radius, and the median, you should pay attention to the property of the circle circumscribed around a triangle with a right angle, which says that the radius is equal to the median drawn to the hypotenuse.Using these properties, the problem is solved in the same manner.

If the question is how to find the hypotenuse of an isosceles right triangle, it is necessary to contact all to the same Pythagorean theorem.But first we recall that the isosceles triangle is a triangle having two equal sides.In the case of a right triangle with sides are the legs of the same.We have FB2 = BK2 + KF2, but as 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, to solve the problem for which you need to calculate the length of the hypotenuse, veryjust.If all of the properties difficult to remember, learn ready-made formulas by substituting known values ​​that can calculate the required length of the hypotenuse.