There is no math exam that does not include the calculation of the hypotenuse of at least one right triangle; however, you don't have to worry as this is a simple calculation! All right-angled triangles have a right angle (90 °) and the side opposite this angle is called the hypotenuse. The Greek philosopher and mathematician Pythagoras, 2500 years ago, found a simple method to calculate the length of this side, which is still used today. This article will teach you to use the 'Pythagorean Theorem' when you know the length of the two legs and use the 'Sine Theorem' when you only know the length of one side and the width of an angle (in addition to the right one). Finally, you will be offered how to recognize and memorize the value of the hypotenuse in special right-angled triangles that often appear in math tests.
Steps
Method 1 of 3: Pythagorean theorem
Step 1. Learn the 'Pythagorean Theorem'
This law describes the relationship between the sides of a right triangle and is one of the most used in mathematics (even in classwork!). The theorem states that in every right triangle whose hypotenuse is 'c' and the legs are 'a' and 'b' the relation holds: to2 + b2 = c2.
Step 2. Make sure the triangle is right
In fact, the Pythagorean Theorem is valid only for this type of triangle, since by definition it is the only one to have a hypotenuse. If the triangle in question has an angle that measures exactly 90 °, then you are facing a right triangle and you can proceed with the calculations.
Right angles are often identified, both in textbooks and in class assignments, with a small square. This special sign means "90 °"
Step 3. Assign the variables a, b and c to the sides of the triangle
The variable "c" is always assigned to the hypotenuse, the longest side. The legs will be a and b (no matter in what order, the result does not change). At this point enter the values corresponding to the variables in the form of the Pythagorean Theorem. For instance:
If the legs of the triangle measure 3 and 4, then assign these values to the letters: a = 3 and b = 4; the equation can be rewritten as: 32 + 42 = c2.
Step 4. Find the squares of a and b
To do this, simply multiply each value by itself, then: to2 = a x a. Find the squares of a and b and enter the results into the formula.
- If a = 3, a2 = 3 x 3 = 9. If b = 4, b2 = 4 x 4 = 16.
- Once these numbers have been entered in the formula, the equation should look like this: 9 + 16 = c2.
Step 5. Add the values of a together2 And b2.
Enter the result in the formula and you will have the value of c2. Only one last step is missing and you will have solved the problem.
In our example you will get 9 + 16 = 25, so you can state that 25 = c2.
Step 6. Extract the square root of c2.
You can use your calculator function (or your memory or multiplication tables) to find the square root of c2. The result corresponds to the length of the hypotenuse.
To finish the calculations of our example: c2 = 25. The square root of 25 is 5 (5 x 5 = 25, so Sqrt (25) = 5). This means that c = 5, the length of the hypotenuse!
Method 2 of 3: Special Triangles Rectangles
Step 1. Learn to recognize the Pythagorean triples
These are composed of three integers (associated with the sides of the right triangles) that satisfy the Pythagorean Theorem. These are triangles that are used very often in geometry textbooks and in class assignments. If you memorize, in particular, the first two Pythagorean triples, you will save a lot of time during the exams because you will immediately know the value of the hypotenuse!
- The first Pythagorean Terna is: 3-4-5 (32 + 42 = 52, 9 + 16 = 25). If you are offered a right triangle whose sides are 3 and 4, you can be sure that the hypotenuse is equal to 5 without having to do any calculations.
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The Pythagorean Terna is also valid for multiples of 3-4-5, as long as the proportions between the various sides are maintained. For example, a right-angled triangle on its side
Step 6
Step 8. will have the even hypotenuse
Step 10. (62 + 82 = 102, 36 + 64 = 100). The same goes for 9-12-15 and also for 1, 5-2-2, 5. Try to verify this yourself with math calculations.
- The second very popular Pythagorean Terna in mathematics exams is 5-12-13 (52 + 122 = 132, 25 + 144 = 169). Also in this case the multiples that respect the proportions are valid, for example: 10-24-26 And 2, 5-6-6, 5.
Step 2. Memorize the ratios between the sides of a triangle with 45-45-90 angles
In this case we are faced with an isosceles right triangle, which is often used in class assignments, and the problems related to it are simple to solve. The relationship between the sides, in this specific case, is 1: 1: Sqrt (2) which means that the cathets are equal to each other and that the hypotenuse is equal to the length of the cathetus multiplied by the root of two.
- To calculate the hypotenuse of an isosceles right triangle of which you know the length of a cathetus, just multiply the latter by the value of Sqrt (2).
- Knowing the ratios between the sides is very useful when the problem gives you the values of the sides expressed as variables and not as integers.
Step 3. Learn the relationship between the sides of a triangle with 30-60-90 angles
In this case you have a right triangle with angles of 30 °, 60 ° and 90 ° which corresponds to one half of an equilateral triangle. The sides of this triangle have a ratio equal to: 1: Sqrt (3): 2 or: x: Sqrt (3) x: 2x. If you know the length of a catheter and you need to find the hypotenuse, the procedure is very simple:
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If you know the value of the minor cathetus (the one opposite the angle of 30 °) simply multiply the length by two and find the value of the hypotenuse. For example, if the minor cathetus is equal to
Step 4., the hypotenuse is the same
Step 8..
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If you know the value of the larger cathetus (the one opposite the 60 ° angle) then multiply its length by 2 / Sqrt (3) and you will get the value of the hypotenuse. For example, if the cathetus is greater
Step 4., the hypotenuse must be 4, 62.
Method 3 of 3: Sine Theorem
Step 1. Understand what "breast" is
The terms "sine," "cosine" and "tangent" all refer to various ratios between the angles and / or sides of a right triangle. In a right triangle the otherwise of an angle is defined as the length of the side opposite the corner divided by the length of the hypotenuse of the triangle. In calculators and equations this function is abbreviated with the symbol: sin.
Step 2. Learn to calculate the sine
Even the simplest scientific calculators have the breast calculation function. Check the key indicated with the symbol sin. To find the sine of an angle, you have to press the key sin and then type the angle value expressed in degrees. In some calculator models, you have to do exactly the opposite. Try some tests or check your calculator manual to understand how it works.
- To find the sine of an angle of 80 °, you have to type since 80 and press the enter key or equal or you have to type 80 left. (The result is -0.9939.)
- You can also do an online search for the words "breast calculator", you will find many virtual calculators that will shed light on many doubts.
Step 3. Learn the 'Sine Theorem'
This is a very useful tool for solving problems related to right triangles. In particular, it allows you to find the value of the hypotenuse when you know the length of one side and the value of another angle in addition to the right one. In any right triangle whose sides are to, b And c with corners TO, B. And C. the Sines Theorem states that: a / sin A = b / sin B = c / sin C.
The Sine Theorem can be applied to solve problems of any triangle, but only the right-angled ones have the hypotenuse
Step 4. Assign the variables a, b and c to the sides of the triangle
The hypotenuse must be "c". For simplicity we call the known side "a" and the other "b". Now assign the variables A, B and C to the corners. The one opposite to the hypotenuse must be called "C". The one opposite side "a" is the angle "A" and the one opposite side "b" is called "B".
Step 5. Calculate the value of the third angle
Since one is righteous, you know that C = 90 ° you can easily calculate the values of TO or B.. The sum of the internal angles of a triangle is always 180 ° so you can set the equation: 180 - (90 + A) = B. which can also be written as: 180 - (90 + B) = A.
For example, if you know that A = 40 °, so B = 180 - (90 + 40). Carrying out the calculations: B = 180 - 130 you get that: B = 50 °.
Step 6. Examine the triangle
At this point you should know the value of the three angles and the length of side a. Now you need to enter this information into the Sine Theorem formula to determine the length of the other two sides.
To continue with our example, consider that a = 10. The angle C = 90 °, the angle A = 40 ° and the angle B = 50 °
Step 7. Apply the Sine Theorem to the triangle
You must enter the known values in the formula and solve it for c (the length of the hypotenuse): a / sin A = c / sin C. The formula may seem complicated but the sine of 90 ° is a constant and is always equal to 1! Now simplify the equation: a / sin A = c / 1 or: a / sin A = c.
Step 8. Divide the length of side a for the sine of the angle A to find the value of the hypotenuse!
You can do this in two different steps, first by calculating the sine of A and noting the result and then dividing the latter by a. Alternatively, enter all values into the calculator. If you prefer this second method, don't forget to type the parentheses after the division sign. For example type: 10 / (sin 40) or 10 / (40 left), based on the calculator model.
