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- Geometry for Beginners – How to Use Pythagorean Triples
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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 list triplets that should be memorized. Knowing Pythagorean triangles can save you so much time and effort when working with right triangles!

In another article, Geometry for Beginners, 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, 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.

This famous theorem has two important applications: (1) to determine whether a triangle is a right triangle when the lengths of all three sides are given, and (2) to find the length of the missing side of a right triangle when the other two sides are known. This second application sometimes creates a Pythagorean triple, a very special set of three numbers.

A Pythagorean triple is a set of three numbers that has two properties: (1) they are **sides of a right triangle**and (2) they are **all integers**. The quality of the integers is particularly important. Since the Pythagorean theorem involves squaring each variable, the process of solving for one variable involves taking the square root of both sides of the equation. Only a few times will “taking the square root” give a whole number. Usually, the missing value is irrational.

For example: **Find the length of the side of a right triangle whose hypotenuse is 8 inches and leg 3 inches**.

**The solution.** Using and remembering the Pythagorean ratio *c* used during the hypotenuse *a* and *b* has 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 *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 *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 whose legs are 6 inches and 8 inches.**

**The solution. **Again, using the Pythagorean theorem, *c*^2 = *a*^2 +* b*^2 changes *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. So the hypotenuse is 10 inches long. WOW! All three sides – 6, 8 and 10 – are whole numbers. This is SPECIAL! These “special” situations are Pythagorean triples.

Pythagorean triples should be considered “families” based on the smallest number in that family. Since the common factor of 6, 8, and 10 is 2, removing that common factor yields the values 3, 4, and 5. When testing 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 sides 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 (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. all Pythagorean triples are in the family 3, 4, 5.

**ATTENTION ALL STUDENTS!** Standardized test writers often use Pythagorean relations in their math questions, so the most commonly used values will be useful to you. However, you must be aware that these same test writers often create questions to confuse those whose understanding of the concept is not quite what it should be.

**Example of the question “Meant to hit you”:** Find the hypotenuse of a right triangle whose legs are 30 and 50 units. The tricky part is that students see a multiplier of 10 and think they have a triplet of 3, 4, 5 with a hypotenuse of 40 units. WRONG! Do you see why this is wrong? You’re not alone if you can’t see it. Remember that the hypotenuse must be the LONGEST side, so 40 cannot be the hypotenuse. Always THINK carefully before jumping to an answer that seems too easy. (Since the triple doesn’t really work here, you have to do the whole formula to find the missing value.)

**Pythagorean triples for memorization and recognition:**

(1) 3, 4, 5 and all its multiples

(2) 5, 12, 13 and all multiples thereof

(3) 8, 15, 17 and all multiples thereof

(4) 7, 24, 25 and all multiples thereof

Memorizing ALL the multiples would be impossible, but you should learn the most commonly used multiples: 2, 3, 4, 5, and 10. The time you save in the years ahead will be worth every moment you spend learning these combinations. !

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