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Geometry for Beginners – How to Use Pythagorean Triples
Welcome to Geometry for Beginners. In this article we will review the Pythagorean Theorem, look at the meaning of the phrase “Pythagorean Triple,” and discuss how these triples are used. In addition, we will list the triples that should be memorized. Knowing Pythagorean Triples can save so much time and effort when working with right triangles!
In another Geometry for Beginners article, we discussed the Pythagorean Theorem. This theorem states a relationship about right triangles that is ALWAYS TRUE: In all right triangles, the square of the hypotenuse is equal to the sum of the squares of the legs. In symbols, this looks like c^2 = a^2 + b^2. This formula is one of the most important and widely used in all of mathematics, so it is important that students understand it uses.
There are two important applications of this famous theorem: (1) to determine if a triangle is a right triangle if the lengths of all 3 sides are given, and (2) to find the length of a missing side of a right triangle if the other two sides are known. This second application sometimes produces a Pythagorean Triple–a very special set of three numbers.
A Pythagorean Triple is a set of three numbers that share two qualities: (1) they are the sides of a right triangle, and (2) they are all integers. The integer quality is especially important. Since the Pythagorean Theorem involves squaring each variable, the process of solving for one of the variables involves taking the square root of both sides of the equation. Only a few times does “taking a square root” produce an integer value. Usually, the missing value will be irrational.
As an example: Find the length of the side of a right triangle with a hypotenuse of 8 inches and a leg of 3 inches.
Solution. Using the Pythagorean relationship and remembering that c is used for the hypotenuse while a and b are the two legs: c^2 = a^2 + b^2 becomes 8^2 = 3^2 + b^2 or 64 = 9 + b^2 or b^2 = 55. To solve for b, we must take the square root of both sides of the equation. Since 55 is NOT a perfect square, we cannot eliminate the radical sign, so b = sqrt(55). This means that the missing length is an irrational number. THIS is a typical result.
This next example is NOT so typical: Find the hypotenuse of a right triangle with legs of 6 inches and 8 inches.
Solution. Again, using the Pythagorean Theorem, c^2 = a^2 + b^2 becomes c^2 = 6^2 + 8^2 or c^2 = 36 + 64 or c^2 = 100. Remember that, algebraically, c has two possible values: +10 and -10; but, geometrically, length cannot be negative. Thus, the hypotenuse is of length 10 inches. WOW! All three sides–6, 8, and 10–are integers. This is SPECIAL! These “special” situations are Pythagorean Triples.
Pythagorean Triples should be considered to occur as “families” based on the smallest set of numbers in that family. Since 6, 8, and 10 have a common factor of 2, removing that common factor results in values of 3, 4, and 5. Testing with the Pythagorean Theorem, we want to know IF 5^2 is equal to 3^2 + 4^2. Is it? Is 25 = 9 + 16? YES! This means that sides of 3, 4, and 5 form a right triangle; and since all values are integers, 3, 4, 5 is a Pythagorean Triple. Thus, 3, 4, 5 and its multiples–like 6, 8, 10 (a multiple of 2) or 9, 12, 15 (a multiple of 3) or 15, 20, 25 (a multiple of 5) or 30, 40, 50 (a multiple of 10), etc., are all Pythagorean Triples in the 3, 4, 5 family.
ATTENTION ALL STUDENTS! The writers of standardized tests often use Pythagorean relationships in their math questions, so it will benefit you to have the most commonly used values memorized. However, you must be aware that these same test writers often construct questions to confuse those whose understanding of the concept is not quite what it should be.
Example of an “intended to catch you” question: Find the hypotenuse of a right triangle having legs of 30 and 50 units. The tricky part is that students see a multiplier of 10 and think they have a 3, 4, 5 triple with a hypotenuse of 40 units. WRONG! Do you see why this is wrong? You won’t be alone if you don’t see it. Remember that the hypotenuse must be the LONGEST side, so 40 cannot be the hypotenuse. Always THINK carefully before jumping on an answer that seems too easy. (Since the triple doesn’t really work here, you will need to do the entire formula to find the missing value.)
Pythagorean Triples to memorize and recognize:
(1) 3, 4, 5 and all of its multiples
(2) 5, 12, 13 and all of its multiples
(3) 8, 15, 17 and all of its multiples
(4) 7, 24, 25 and all of its multiples
Memorizing ALL of the multiples would be impossible, but you should learn the most commonly used multipliers: 2, 3, 4, 5, and 10. The time you will save in the years ahead is worth every minute you spend now to learn these combination’s!
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