Geometry for Beginners - How to Use Pythagorean Triples

Free Math Worksheets - Geometry for Beginners - How to Use Pythagorean Triples

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Welcome to Geometry for Beginners. In this narrative we will relate 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 triples that should be memorized. Knowing Pythagorean Triples can save so much time and endeavor when working with right triangles!

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In an additional one Geometry for Beginners article, we discussed the Pythagorean Theorem. This theorem states a relationship about right triangles that is all the time True: In all right triangles, the quadrate 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 method is one of the most leading and widely used in all of mathematics, so it is leading that students understand it uses.

There are two leading applications of this supreme theorem: (1) to conclude if a triangle is a right triangle if the lengths of all 3 sides are given, and (2) to find the distance 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 extra 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 triangle, and (2) they are all integers. The integer quality is especially important. Since the Pythagorean Theorem involves squaring each variable, the process of solving for one of the variables involves taking the quadrate root of both sides of the equation. Only a few times does "taking a quadrate root" furnish an integer value. Usually, the missing value will be irrational.

As an example: Find the distance of the side of a right triangle with a hypotenuse of 8 inches and a leg of 3 inches.

Solution. Using the Pythagorean relationship and remembering that c is used for the hypotenuse while a and b are the 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 for b, we must take the quadrate root of both sides of the equation. Since 55 is Not a excellent square, we cannot eliminate the radical sign, so b = sqrt(55). This means that the missing distance is an irrational number. This is a typical result.

This next example is Not so typical: Find the hypotenuse of a right triangle with legs of 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 or c^2 = 100. Remember that, algebraically, c has two possible values: +10 and -10; but, geometrically, distance cannot be negative. Thus, the hypotenuse is of distance 10 inches. Wow! All three sides--6, 8, and 10--are integers. This is Special! These "special" situations are Pythagorean Triples.

Pythagorean Triples should be considered to occur as "families" based on the smallest set of numbers in that family. Since 6, 8, and 10 have a base factor of 2, removing that base 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? Is 25 = 9 + 16? Yes! This means that 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 (a complicated of 2) or 9, 12, 15 (a complicated of 3) or 15, 20, 25 (a complicated of 5) or 30, 40, 50 (a complicated of 10), etc., are all Pythagorean Triples in the 3, 4, 5 family.

Attention All Students! The writers of standardized tests often use Pythagorean relationships in their math questions, so it will advantage you to have the most generally used values memorized. However, you must be aware that these same test writers often invent questions to confuse those whose comprehension of the idea is not quite what it should be.

Example of an "intended to catch you" question: Find the hypotenuse of a right triangle having legs of 30 and 50 units. The tricky part is that students see a multiplier 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? 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. all the time Think considered before jumping on an write back that seems too easy. (Since the triple doesn't authentically work here, you will need to do the entire method to find the missing value.)

Pythagorean Triples to memorize and recognize:

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

(2) 5, 12, 13 and all of its multiples

(3) 8, 15, 17 and all of its multiples

(4) 7, 24, 25 and all of its multiples

Memorizing All of the multiples would be impossible, but you should learn the most generally used multipliers: 2, 3, 4, 5, and 10. The time you will save in the years ahead is worth every little you spend now to learn these combination's!

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