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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 triples” and discuss how these triples are used. In addition, we will list the triads that must be memorized. Knowing Pythagorean triangles 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 states a relationship that is ALWAYS TRUE about right triangles: 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 mathematics, so it is important for students to use it.

There are two important applications of this famous theorem: (1) to determine whether 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. two other aspects 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 properties: (1) they are **sides of a right triangle**and (2) they are **all numbers**. The quality of the number 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 the square root” produce an integer value. In general, the lost value will be negligible.

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

**Solution.** Use and remember the Pythagorean relation *c* When used for the hypotenuse *one* and *b* have two legs: *c*^2 = *one*^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, so *b* = sqrt (55). This means that the length is missing or e *silly* the number THIS is a typical result.

This next example 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 = *one*^2 +* b*^2 is possible *c*^2 = 6^2 + 8^2 or* c*^2 = 36 + 64 or *c*^2 = 100. Remember that, algebraically, *c* there are two possible values: +10 and -10; but, geometrically, length cannot be negative. Thus, the hypotenuse is 10 inches long. WOW! All three sides — 6, 8 and 10 — are numbers. This is EXCLUSIVE! These “special” cases are Pythagorean Triples.

Pythagorean triples should be considered a “family” based on the smallest number of numbers in that family. Since 6, 8, and 10 have a common factor of 2, eliminating the common factor yields the values 3, 4, and 5. By testing 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 3, 4, and 5 form a right triangle; and since all values are numbers, 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., all Pythagorean triples are in 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 memorize the most frequently used values. However, you should know that these same test writers often create questions to confuse those whose understanding is not what it should be.

**An example of a “what do you mean” question:** Find the hypotenuse of a right triangle whose legs are 30 and 50 units. The hard part is when students see a multiple of 10 and think that three, 3, 4, 5 has a hypotenuse of 40 units. FAKE! Do you see why this is wrong? If you don’t see it you won’t be alone. Remember that the hypotenuse must be the LONGEST side, so 40 cannot be the hypotenuse. Always do your due diligence before jumping on an answer that seems too easy. (Since the third doesn’t really work here, you’ll need to do the whole formula to find the missing value.)

**Pythagorean triples multiplication and recognition:**

(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 numbers will be impossible, but you should learn the most commonly used numbers: 2, 3, 4, 5, and 10. The time you’ll save in the years to come is worth every minute you spend now. to learn these combinations. !

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