Is The Hypotenuse Always C

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All correct triangles have one right (90-degree) bending, and the hypotenuse is the side that is opposite or the correct bending, or the longest side of the right triangle.^[ane] The hypotenuse is the longest side of the triangle, and it's also very like shooting fish in a barrel to detect using a couple of different methods. This article will teach you how to find the length of the hypotenuse using the Pythagorean theorem when yous know the length of the other two sides of the triangle. Information technology will and so teach you to recognize the hypotenuse of some special right triangles that oftentimes appear on tests. Information technology volition finally teach you to notice the length of the hypotenuse using the Law of Sines when yous simply know the length of i side and the measure of i additional angle.

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Learn the Pythagorean Theorem. The Pythagorean Theorem describes the relationship betwixt the sides of a correct triangle.^[2] Information technology states that for any right triangle with sides of length a and b, and hypotenuse of length c, aⁱⁱ + b² = c² .^[3]
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Make sure that your triangle is a right triangle. The Pythagorean Theorem simply works on correct triangles, and past definition only right triangles can have a hypotenuse. If your triangle contains one angle that is exactly xc degrees, it is a right triangle and you tin can proceed.
- Right angles are oftentimes notated in textbooks and on tests with a pocket-sized square in the corner of the angle. This special mark means "90 degrees."
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Assign variables a, b, and c to the sides of your triangle. The variable "c" will always exist assigned to the hypotenuse, or longest side. Cull 1 of the other sides to exist a, and call the other side b (information technology doesn't affair which is which; the math volition plow out the aforementioned). So copy the lengths of a and b into the formula, according to the post-obit example:
- If your triangle has sides of 3 and 4, and you lot have assigned letters to those sides such that a = 3 and b = 4, then yous should write your equation out as: 3^two + 4² = c² .
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Find the squares of a and b. To notice the square of a number, you simply multiply the number by itself, so a^two = a 10 a. Discover the squares of both a and b, and write them into your formula.^[four]
- If a = 3, a^two = 3 x 3, or ix. If b = 4, then b^two = 4 x 4, or 16.
- When y'all plug those values into your equation, information technology should now expect similar this: 9 + 16 = c² .
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Add together together the values of aⁱⁱ and b^two . Enter this into your equation, and this will give you the value for c². There is merely one stride left to go, and yous will have that hypotenuse solved!
- In our example, nine + 16 = 25, so you should write downwards 25 = c² .
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Notice the foursquare root of c². Use the square root part on your computer (or your memory of the multiplication tabular array) to discover the square root of c². The reply is the length of your hypotenuse!^[5]
- In our case, c² = 25. The square root of 25 is 5 (v 10 5 = 25, then Sqrt(25) = v). That means c = 5, the length of our hypotenuse!

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Larn to recognize Pythagorean Triple Triangles. The side lengths of a Pythagorean triple are integers that fit the Pythagorean Theorem. These special triangles appear frequently in geometry text books and on standardized tests similar the Sabbatum and the GRE. If y'all memorize the first 2 Pythagorean triples, in particular, you lot can save yourself a lot of fourth dimension on these tests because you tin can immediately know the hypotenuse of one of these triangles just by looking at the side lengths!^[6]
- The commencement Pythagorean triple is 3-4-5 (3² + ivⁱⁱ = 5ⁱⁱ, 9 + 16 = 25). When you see a right triangle with legs of length 3 and iv, you lot can instantly be certain that the hypotenuse will exist five without having to do any calculations.
- The ratio of a Pythagorean triple holds true even when the sides are multiplied by another number. For case a correct triangle with legs of length six and eight volition take a hypotenuse of ten (half-dozen² + 8^two = ten², 36 + 64 = 100). The aforementioned holds true for 9-12-15, and fifty-fifty one.5-2-2.5. Attempt the math and see for yourself!
- The 2nd Pythagorean triple that commonly appears on tests is v-12-13 (five² + 12² = 13^two, 25 + 144 = 169). Also be on the lookout for multiples like 10-24-26 and ii.5-6-6.v.
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Memorize the side ratios of a 45-45-90 correct triangle. A 45-45-xc right triangle has angles of 45, 45, and 90 degrees, and is also chosen an Isosceles Right Triangle. It occurs frequently on standardized tests, and is a very easy triangle to solve. The ratio between the sides of this triangle is one:1:Sqrt(2), which means that the length of the legs are equal, and the length of the hypotenuse is simply the leg length multiplied past the square root of two.^[7]
- To calculate the hypotenuse of this triangle based on the length of one of the legs, only multiply the leg length past Sqrt(2).
- Knowing this ratio comes in especially handy when your exam or homework question gives yous the side lengths in terms of variables instead of integers.
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Learn the side ratios of a thirty-lx-90 right triangle. This triangle has angle measurements of xxx, 60, and xc degrees, and occurs when you cut an equilateral triangle in half. The sides of the 30-60-90 right triangle always maintain the ratio one:Sqrt(3):2, or x:Sqrt(three)ten:2x. If y'all are given the length of 1 leg of thirty-60-90 right triangle and are asked to find the hypotenuse, it is very easy to practise:^[8]
- If you are given the length of the shortest leg (contrary the 30-caste angle,) simply multiply the leg length by 2 to find the length of the hypotenuse. For example, if the length of the shortest leg is iv, you know that the hypotenuse length must be eight.
- If you are given the length of the longer leg (opposite the lx-degree bending,) multiply that length by two/Sqrt(3) to find the length of the hypotenuse. For instance, if the length of the longer leg is four, y'all know that the hypotenuse length must be 4.62.

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Understand what "Sine" means. The terms "sine," "cosine," and "tangent" all refer to various ratios betwixt the angles and/or sides of a correct triangle. In a right triangle, the sine of an angle is defined as the length of the side contrary the angle divided by the hypotenuse of the triangle. The abbreviation for sine found in equations and on calculators is sin .^[ix]
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Learn to summate sine. Fifty-fifty a basic scientific calculator will have a sine role. Look for a fundamental marked sin . To detect the sine of angle, you will usually press the sin key and then enter the bending measurement in degrees. On some calculators, however, you must enter the degree measurement first and and so the sin key. Y'all volition take to experiment with your calculator or check the manual to find out which it is.
- To find the sine of an 80 caste angle, you will either need to key in sin 80 followed by the equal sign or enter cardinal, or 80 sin . (The answer is -0.9939.)
- Yous tin besides blazon in "sine figurer" into a web search, and detect a number of like shooting fish in a barrel-to-use calculators that volition remove any guesswork.^[ten]
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Learn the Law of Sines. The Law of Sines is a useful tool for solving triangles. In detail, it can help you find the hypotenuse of a correct triangle if you know the length of one side, and the measure of i other angle in addition to the correct angle. For whatsoever triangle with sides a, b, and c, and angles A, B, and C, the Constabulary of Sines states that a / sin A = b / sin B = c / sin C.^[11]
- The Police force of Sines can actually be used to solve any triangle, but only a right triangle will have a hypotenuse.
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Assign the variables a, b, and c to the sides of your triangle. The hypotenuse (longest side) must be "c". For the sake of simplicity, characterization the side with the known length equally "a," and the other "b". Then assign variables A, B, and C to the angles of the triangle. The right angle opposite the hypotenuse will be "C". The angle reverse side "a" is bending "A," and the angle opposite side "b" is "B".
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Calculate the measurement of the 3rd angle. Because information technology is a right angle, you already know that C = 90 degrees, and you as well know the measure out of A or B. Since the internal degree measurement of a triangle must always equal 180 degrees, you can easily calculate the measurement of the third angle using the following formula: 180 – (90 + A) = B. You can also reverse the equation such that 180 – (90 + B) = A.
- For example, if y'all know that A = 40 degrees, and then B = 180 – (ninety + 40). Simplify this to B = 180 – 130, and you tin can quickly decide that B = l degrees.
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Examine your triangle. At this point, you should know the caste measurements of all three angles, and the length of side a. It is at present time to plug this information into the Law of Sines equation to determine the lengths of the other two sides.
- To go along our example, let's say that the length of side a = 10. Angle C = 90 degrees, angle A = forty degrees, and angle B = fifty degrees.
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Apply the Law of Sines to your triangle. We just need to plug our numbers in and solve the following equation to determine the length of hypotenuse c: length of side a / sin A = length of side c / sin C. This might nonetheless look a bit intimidating, but the sine of 90 degrees is a constant, and ever equals i! Our equation can thus be simplified to: a / sin A = c / 1, or only a / sin A = c.^[12]
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Dissever the length of side a by the sine of angle A to find the length of the hypotenuse! You can do this in two split steps, by get-go calculating sin A and writing information technology downwards, and and then dividing past a. Or you can cardinal information technology all into the calculator at the aforementioned time. If yous practise, remember to include parentheses after the sectionalisation sign.^[xiii] For instance, fundamental in either ten / (sin forty) or 10 / (40 sin), depending on your calculator.
- Using our case, we find that sin 40 = 0.64278761. To discover the value of c, we simply split the length of a past this number, and learn that 10 / 0.64278761 = 15.half-dozen, the length of our hypotenuse!

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How exercise you notice the length of the hypotenuse in the Pythagorean Theorem?

Grace Imson, MA
Math Instructor, City College of San Francisco

Grace Imson is a math teacher with over 40 years of teaching feel. Grace is currently a math instructor at the Metropolis College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the simple, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University.

Math Instructor, City College of San Francisco

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1 mutual error is forgetting to square the terms. In the Pythagorean Theorem, all three terms are squared. Many people go too fast and forget to find the square before the sum of 'a' and 'b,' which gives them an incorrect reply.
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Is there a computer for finding the length of the hypotenuse?

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Google provides a correct angle triangle reckoner that allows you to solve for the hypotenuse. Simply search for "hypotenuse calculator" and plug your numbers into the calculator at the summit of the search results. You lot tin as well use the hypotenuse estimator at Omincalculator.com.
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How tin you find the length of the hypotenuse given the length of i side and an angle?

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If y'all know yous are dealing with a right triangle, and so you already know that one of the angles is 90°. Since the angles must add up to 180°, you tin solve for the missing bending using the formula 90 + 10 = 180. Once you have all three angles, you can use that information and the known length of 1 side to use the law of sines and find the length of the hypotenuse.

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If you lot need to find the length of the hypotenuse of a right triangle, you can use the Pythagorean theorem if you lot know the length of the other 2 sides. Square the length of the 2 sides, called a and b, and so add together them together. Take the square root of the outcome to get the hypotenuse. If y'all want to learn how to find the hypotenuse using trigonometric functions, keep reading the article!

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