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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 triples that should be remembered. 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 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 its use.

This famous theorem has two important uses: (1) to determine whether a triangle is a right triangle given the lengths of all 3 sides, and (2) to find the length of the 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 properties: (1) They are **The sides of a right triangle**and (2) they are **All integers**. Integer quality is especially important. Since the Pythagorean theorem squares each variable, the process of solving for one variable 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 missing value will be irrational.

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

**Solution.** Using and remembering the Pythagorean relation *c* while is used for karna *a* and *b* The two legs are: *c*^2 = *a*^ 2 + *b*^2 becomes 8^2 = 3^2 +* b*^2 or 64 = 9 + *b*^2 or *b*^2 = 55. to resolve *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* = sqrt(55). This means the missing length is one *irrational* the number This is a normal result.

This next example is not very common: **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*becomes ^2 *c*^2 = 6^2 + 8^2 Or* c*^2 = 36 + 64 Or *c*^2 = 100. Note that, algebraically, *c* There are two possible values: +10 and -10; But, geometrically, length cannot be negative. Thus the length of the hypotenuse is 10 inches. Wow! All three sides—6, 8, and 10—are integers. This is special! These “special” conditions are Pythagorean triples.

Pythagorean Triples appear as “families” based on a small set of numbers in that family. Since 6, 8, and 10 have a common factor of 2, eliminating that common factor yields 3, 4, and 5. Testing with the Pythagorean Theorem, we want to know IF 5^2 equals 3^2 + 4^2. yes Is 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. Thus, 3, 4, 5 and its multiples – like 6, 8, 10 (multiplication of 2) or 9, 12, 15 (multiplication of 3) or 15, 20, 25 (multiplication of 5) or 30 , 40, 50 (multiples of 10), etc., 3, 4, 5 are all Pythagorean triples in the family.

**Attention all students!** The writers of standardized tests often use Pythagorean relations in their math questions, so it will benefit you to memorize the most frequently used values. However, you should be aware that these same test writers often construct questions whose understanding of concepts is not entirely as it should be.

**Example of an “intention to catch you” question:** Find the hypotenuse of a right triangle with legs of 30 and 50 units. The tricky part is that students see a multiple 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? If you haven’t seen it, you’re not alone. Note that the hypotenuse must be the longest side, so there cannot be 40 hypotenuses. Always think carefully before jumping to an answer that seems too easy. (Since triples don’t really work here, you have to do the whole formula to find the missing value.)

**To remember and identify Pythagorean triples:**

(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

It will be impossible to memorize all the multiples, but you should learn the most frequently used multiples: 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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