A Pythagorean triple is a set of three whole numbers that can be the side lengths of a right triangle. Substitute the given numbers into the Pythagorean theorem to see whether or not they make a Pythagorean triple. (? = yes; ? = no) Show your work. The first one is done for you. 1. a = 9; b = 40; c = 42 9² + 40² ? 42²; 81 + 1600 = 1681; 42² = 1764; 1681 ? 1764 ? 2. a = 7; b = 24; c = 25 3. a = 11; b = 59; c = 60 4. Discover another Pythagorean triple by taking one of these sets that works and multiplying each number by any positive integer. Show that your new set of numbers works in the Pythagorean Theorem.
Added by Steven B.
Close
Step 1
Now, let's multiply each number by a positive integer. For simplicity, let's choose 2. So, our new set of numbers is (a=14, b=48, c=50). Now, let's substitute these numbers into the Pythagorean theorem to see if they still work: Show more…
Show all steps
Your feedback will help us improve your experience
Sri K and 76 other Geometry educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Pythagorean triples, denoted by $(a, b, c),$ represent three positive integers $a, b,$ and $c$ that satisfy the relationship $a^{2}+b^{2}=c^{2}$. Suppose that $a=4$. To find all values of $b$ and $c$ that would form Pythagorean triples with $a=4$ we can use the following technique: Substituting $a=4$, we have $4^{2}+b^{2}=c^{2}$. Therefore, $c^{2}-b^{2}=16 .$ In factored form this is $(c-b)(c+b)=16$. Setting $c-b$ and $c+b$ equal to the positive factors of 16 and solving the resulting system of equations, we have $$ \begin{array}{lllll} c-b=1 & c-b=2 & c-b=4 & c-b=8 & c-b=16 \\ c+b=16 & c+b=8 & c+b=4 & c+b=2 & c+b=1 \end{array} $$ The only system that returns positive integer values for $b$ and $c$ is the second system. It gives $c=5$ and $b=3$ resulting in a Pythagorean triple of (4,3,5) . Use this technique for Exercises $93-94$ Find all Pythagorean triples $(a, b, c)$ such that $b=12$.
Systems of Equations and Inequalities
Systems of Linear Equations in Two Variables and Applications
Use the fact that a Pythagorean triple is a group of three integers, such as 3, 4, and 5, that could be the lengths of the sides of a right triangle. Find two other Pythagorean triples that are not multiples of 3, 4, 5 or of each other.
Radicals and Connections co Geometry
The Pythagorean Theorem and Its Converse
Recommended Textbooks
Geometry A Common Core Curriculum
Geometry
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD