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Geometry for Beginners – How to Use Pythagorean Triples
Welcome to Geometry for Beginners. In this article, we’ll review the Pythagorean theorem, examine the meaning of the phrase “Pythagorean triple”, and discuss how these triples are used. Additionally, we will list the triplets that need to be memorized. Knowing the Pythagorean triples can save you a lot of time and effort when working with right triangles!
In another article on geometry for beginners, we discussed the Pythagorean theorem. This theorem states a relation on right triangles which 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, it 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 its use.
There are two important applications of this famous theorem: (1) to determine if a triangle is a right triangle if the lengths of the 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 triangleand (2) they are all integers. The whole quality is particularly important. Since the Pythagorean theorem involves squaring each variable, the process of solving any of the variables involves taking the square root of both sides of the equation. Only a few times “taking a square root” produces an integer value. Usually the missing value will be irrational.
For example: Find the side length of a right triangle with an 8 inch hypotenuse and a 3 inch leg.
The solution. Use the Pythagorean relation and remember that vs is used for the hypotenuse while a and b are the two legs: vs^2 = a^2 + b^2 becomes 8^2 = 3^2 + b^2 or 64 = 9 + b^2 or b^2 = 55. To solve b, 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 a irrational Number. This is a typical result.
This next example is NOT so typical: Find the hypotenuse of a right triangle with 6 inch and 8 inch legs.
The solution. Again, using the Pythagorean theorem, vs^2 = a^2 + b^2 becomes vs^2 = 6^2 + 8^2 or vs^2 = 36 + 64 or vs^2 = 100. Recall that, algebraically, vs has two possible values: +10 and -10; but, geometrically, the length cannot be negative. Thus, the hypotenuse has a length of 10 inches. WOW! All three sides – 6, 8 and 10 – are integers. It 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, removing this 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? Does 25 = 9 + 16? YES! This means that the sides of 3, 4 and 5 form a right triangle; and since all values are integers, 3, 4, 5 is a Pythagorean triple. So, 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 of the family 3, 4, 5.
ATTENTION ALL STUDENTS! Standardized test writers often use Pythagorean relations in their math questions, so it will benefit you to memorize the most commonly used values. However, you should be aware that these same test writers often construct questions to confuse those whose understanding of the concept isn’t quite what it should be.
Example of a “meant to catch you” question: Find the hypotenuse of a right triangle with sides 30 and 50 units. The tricky part is that students see a multiplier of 10 and think they have a triplet 3, 4, 5 with a hypotenuse of 40 units. FAKE! 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.)
Pythagorean Triplets to memorize and recognize:
(1) 3, 4, 5 and all its multiples
(2) 5, 12, 13 and all its multiples
(3) 8, 15, 17 and all its multiples
(4) 7, 24, 25 and all its 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 years to come is worth every minute you spend learning these combinations now. !
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