Understanding Pythagorean Triplets in Geometry

A Pythagorean triplet is a set of three positive integers, say a, b, and c, such that ${a}^{2} + {b}^{2} = {c}^{2}$. In a right-angled triangle, if a and b are the lengths of the two legs (perpendicular and base), then c is the length of the hypotenuse. The largest number in the set of three integers is always the hypotenuse.

How to Identify a Pythagorean Triplet

To check if a given set of three numbers forms a Pythagorean triplet, identify the largest number and square it. Then, square the other two numbers and add their squares. If the sum of the squares of the two smaller numbers equals the square of the largest number, the set is a Pythagorean triplet.

Analyzing the Given Options for Pythagorean Triplets

Let's test each option provided to see if it satisfies the Pythagorean condition ${a}^{2} + {b}^{2} = {c}^{2}$, where c is the largest number in the set.

Option 1: 7, 9, 12

The largest number is 12. We need to check if ${7}^{2} + {9}^{2} = {12}^{2}$.

• Calculate the squares: ${7}^{2} = 49$, ${9}^{2} = 81$, ${12}^{2} = 144$.
• Add the squares of the two smaller numbers: $49 + 81 = 130$.
• Compare the sum with the square of the largest number: $130 \neq 144$.

Thus, (7, 9, 12) is not a Pythagorean triplet.

Option 2: 5, 12, 13

The largest number is 13. We need to check if ${5}^{2} + {12}^{2} = {13}^{2}$.

• Calculate the squares: ${5}^{2} = 25$, ${12}^{2} = 144$, ${13}^{2} = 169$.
• Add the squares of the two smaller numbers: $25 + 144 = 169$.
• Compare the sum with the square of the largest number: $169 = 169$.

Thus, (5, 12, 13) is a Pythagorean triplet.

Option 3: 2, 3, 4

The largest number is 4. We need to check if ${2}^{2} + {3}^{2} = {4}^{2}$.

• Calculate the squares: ${2}^{2} = 4$, ${3}^{2} = 9$, ${4}^{2} = 16$.
• Add the squares of the two smaller numbers: $4 + 9 = 13$.
• Compare the sum with the square of the largest number: $13 \neq 16$.

Thus, (2, 3, 4) is not a Pythagorean triplet.

Option 4: 17, 21, 27

The largest number is 27. We need to check if ${17}^{2} + {21}^{2} = {27}^{2}$.

• Calculate the squares: ${17}^{2} = 289$, ${21}^{2} = 441$, ${27}^{2} = 729$.
• Add the squares of the two smaller numbers: $289 + 441 = 730$.
• Compare the sum with the square of the largest number: $730 \neq 729$.

Thus, (17, 21, 27) is not a Pythagorean triplet.

Based on the analysis, only the set (5, 12, 13) satisfies the condition for being a Pythagorean triplet.

Revision Table: Key Concepts on Pythagorean Triplets

Additional Information about Pythagorean Triplets and Triangles

Pythagorean triplets are fundamental to understanding right-angled triangles and the Pythagorean theorem. They provide whole number examples for triangle side lengths, making calculations simpler. The most famous Pythagorean triplet is (3, 4, 5). Multiples of Pythagorean triplets are also Pythagorean triplets; for instance, (6, 8, 10) is a multiple of (3, 4, 5) and is also a triplet because ${6}^{2} + {8}^{2} = 36 + 64 = 100 = {10}^{2}$. Studying these triplets helps in various geometry problems and algebraic manipulations.