After studying the topic of rectangular triangles, students often throw out of their heads 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 must be found. Or the diameter of the circle coincides with the largest side of the triangle, one of whose angles is a straight line. And to find it without this knowledge is impossible.
There are several options for how to find the hypotenuse of a triangle. The choice of method depends on the initial data set in the condition of the problem of quantities.
Method number 1: given both legs
This is the most memorable method, because it uses the Pythagorean theorem. Only sometimes students forget that this formula is the square of the hypotenuse. So, to find the very side, you will need to extract the square root. Therefore, the formula for the hypotenuse, which is usually denoted by the letter "c", will look like this:
c = √ (a 2 + in 2). where the letters "a" and "c" are written both legs of a right triangle.
Method number 2: Knows the catheter and the angle that it is subject to
In order to learn how to find the hypotenuse, you need to remember the trigonometric functions. Namely, the cosine. For convenience, we will assume that we are given a catet a and an angle α adjacent to it.
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 be the value of the leg, and the denominator will be the hypotenuse. From this it follows that the latter can be calculated by the formula:
Method number 3: given the cathetus and the angle that lies opposite it
In order not to get confused in the formulas, we introduce the notation for this angle - β, and the side we leave the previous "a". In this case, we need another trigonometric function, the 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:
In order not to get confused in trigonometric functions, you can remember a simple mnemonic pril: if the problem is aboutabout the angle, then it should be used withand nous, if - about prand lying, then toabout sinus. You should pay attention to the first vowels in the keywords. They form pairso-and orand about .
Method number 4: along the radius of the circumscribed circle
Now, in order to learn how to find the hypotenuse, you need to remember the property of the circle, which is described near the right triangle. It reads as follows. The center of the circle coincides with the middle of the hypotenuse. If to say differently, then the largest side of a right-angled triangle is equal to the diagonal of the circle. That is doubled radius. The formula for this task will look like this:
c = 2 * r. where the letter r denotes the known radius.
These are all possible ways of finding the hypotenuse of a right-angled triangle. Use in each specific task you need the method that is more suitable for the data set.
Example of problem # 1
Condition: in the right-angled triangle, medians are drawn to both legs. The length of the one that is drawn to the larger side is √52. The other median has a length of √73. It is required to calculate the hypotenuse.
Since medians are drawn in the triangle, they divide the legs into two equal segments. For the convenience of reasoning and looking for how to find the hypotenuse, you need to enter several notations. Let both halves of the larger leg be indicated by the letter "x", and the other by "y".
Now we need to consider two rectangular triangles, the hypotenuse of which are the known medians. For them, we need to write twice the formula of the Pythagorean theorem:
(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, according to them, its hypotenuse.
First you need to raise everything to the second power. It turns out:
It can be seen from the second equation that y 2 = 73 - 4x 2. This expression must be substituted in the first and calculate "x":
4 (73 - 4х 2) + х 2 = 52.
292 - 16 x 2 + x 2 = 52 or 15 x 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 6 and 8. Hence, we can use the formula from the first method and find the hypotenuse:
√ (6 2 + 8 2) = √ (36 + 64) = √100 = 10.
Answer. the hypotenuse is 10.
Example problem number 2
Condition: calculate the diagonal in the rectangle with the smaller 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 the rectangle is the largest side in the triangle with an angle of 90 °. Therefore, it all comes down to how to find the hypotenuse.
In the problem we are talking about corners. This means that one will need to use one of the formulas in which trigonometric functions are present. And first you need to determine the value of one of the sharp angles.
Let the smaller of the angles referred to in the condition be denoted by α. Then the right angle that divides the diagonal is 3α. The mathematical notation of this is as follows:
90º = 3?.
It is easy to determine α from this equation. It will be equal to 30º. And it will lie opposite the smaller side of the rectangle. Therefore, the formula described in Method 3 is required.
Hypotenuse is equal to the ratio of the leg to the sinus of the opposite angle, that is:
41 / sin 30º = 41 / (0.5) = 82.
Answer: the hypotenuse is 82.