Among the many calculations performed to calculate certain quantities of various geometric shapes, there is a hypotenuse of the triangle. Recall that a triangle is a polyhedron having three angles. Below are several methods for calculating the hypotenuse of various triangles.
First, let's see how to find the hypotenuse of a right triangle. For those who have forgotten, a triangle is called a rectangular, having an angle of 90 degrees. The side of the triangle located on the opposite side of the 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:
- Known lengths of legs. The hypotenuse in this case is calculated using the Pythagorean theorem, which reads as follows: the square of the hypotenuse is equal to the sum of the squares of the legs. If we consider the 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 sizes in turn. Then add the learned numbers and extract the square root from the result.
Consider an example: Given a triangle with a right angle. One leg is 3 cm, the other is 4 cm. Find hypotenuse. The solution is as follows.
FB2 = BK2 + KF2 = (3cm) 2+ (4cm) 2 = 9cm2 + 16cm2 ββ= 25 cm2. We take the square root 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? Denote the known angle Ξ±. According to the property of a right-angled 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 this leg and the hypotenuse. Considering the triangle, this can be written as follows: FB = BK * cos (Ξ±).
- A leg (KF) and the same angle Ξ± are known, only now it will be opposite. How to find hypotenuse in this case? Let us turn to the same properties of a right-angled triangle and find out that the ratio of the length of the leg to the length of the hypotenuse equals the sine of the angle of the opposite leg. That is, FB = KF * sin (Ξ±).
Let's look at an example. All the same right triangle BKF with hypotenuse FB is given. Let the angle F be 30 degrees, the second angle B corresponds to 60 degrees. The leg BK is also known, the length of which corresponds to 8 cm. The desired value can be calculated as follows:
FB = BK / cos60 = 8 cm.
FB = BK / sin30 = 8 cm.
- The radius of a circle (R) described near a triangle with a right angle is known. How to find hypotenuse when considering such a problem? From the property 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 that divides 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 a similar problem is given, in which the median is not known, then one should pay attention to the property of the circle circumscribed around the 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-angled triangle, the sides are the same sides. We have FB2 = BK2 + KF2, but since BK = KF we have the following: FB2 = 2 BK2, FB = BKβ2
As you can see, knowing the Pythagorean theorem and the properties of a right-angled triangle, it is very simple to solve problems for which it is necessary to calculate the length of the hypotenuse. If itβs difficult to remember all the properties, learn the ready-made formulas, substituting into which the known values ββcan calculate the desired length of the hypotenuse.