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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 triplets that must be memorized. Knowing Pythagorean triples can save a lot of time and effort when working with right triangles!
In another Geometry for Beginners article, we discussed the Pythagorean Theorem. This theorem establishes 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 used in all of mathematics, so it is important that students understand its uses.
There are two important applications of this famous theorem: (1) to determine whether a triangle is a right triangle given the lengths of 3 sides, and (2) to find the length of a missing side of a triangle rectangle if two other 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 triangleand (2) they are all whole numbers. The quality of integers 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. Normally, the missing value will be irrational.
As an example: Find the side length of a right triangle with a hypotenuse of 8 inches and a leg of 3 inches.
solution Using the Pythagorean relation and remembering it c is used for the hypotenuse while a i b are the two legs: c^2 = a^2 + b^2 becomes 8^2 = 3^2 + b^2 or 64 = 9 + b^2 o b^2 = 55. To solve b, we must take the square root of both sides of the equation. Since 55 is NOT a perfect square, we cannot remove the radical sign, therefore b = square (55). This means that the missing length is one irrational number THIS is a typical result.
This example below is NOT so typical: Find the hypotenuse of a right triangle with legs 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 o c^2 = 100. Remember that, algebraically, c has two possible values: +10 and -10; but, geometrically, the length cannot be negative. Therefore, the hypotenuse has a length of 10 inches. WOW! All three sides (6, 8 and 10) are whole numbers. This is SPECIAL! These “special” situations are Pythagorean Triples.
Pythagorean triples should be thought of as “families” based on the smallest set of numbers in that family. Since 6, 8, and 10 have a common factor of 2, eliminating that common factor results in values of 3, 4, and 5. Checking with the Pythagorean theorem, we want to know IF 5^2 is equal to 3^2 + 4^2. Is this? Is 25 = 9 + 16? YES! This means that the sides of 3, 4 and 5 form a right triangle; and since all the values are integers, 3, 4, 5 is a Pythagorean triple. So 3, 4, 5 and their multiples such as 6, 8, 10 (multiple of 2) or 9, 12, 15 (multiple of 3) or 15, 20, 25 (multiple of 5) or 30 , 40 , 50 (multiple of 10), etc., are all Pythagorean triples of the 3, 4, 5 family.
ATTENTION ALL STUDENTS! Standardized test writers often use Pythagorean relations in their math questions, so it will benefit you to have the most commonly used values memorized. However, you should be aware that these same test writers often construct questions to confuse those who understand the concept is not what it should be.
Example of a “meant to trap you” question: Find the hypotenuse of a right triangle with legs of 30 and 50 units. The tricky part is that students see a multiplier of 10 and think they have a triple of 3, 4, 5 with a hypotenuse of 40 units. BAD! 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’ll have to do the whole formula to find the missing value.)
Triple Pythagoras to memorize and recognize:
(1) 3, 4, 5 and all their multiples
(2) 5, 12, 13 and all their multiples
(3) 8, 15, 17 and all their multiples
(4) 7, 24, 25 and all their multiples
Memorizing ALL the multiples would be impossible, but you should learn the most commonly used multipliers: 2, 3, 4, 5, and 10. The time you’ll save in the years to come is worth every minute you spend now learning these combinations. !
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