Understanding Pythagorean Triplets

A Pythagorean triplet is a set of three positive integers, say $a$, $b$, and $c$, such that they satisfy the equation $a^2 + b^2 = c^2$. This relationship is derived from the Pythagorean theorem, which applies to right-angled triangles. In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle, which is the longest side) is equal to the sum of the squares of the lengths of the other two sides. If $a$ and $b$ are the lengths of the two shorter sides (legs) and $c$ is the length of the hypotenuse, then $a^2 + b^2 = c^2$.

Identifying the Pythagorean Triplet

To determine which of the given options is a Pythagorean triplet, we need to check if the sum of the squares of the first two numbers equals the square of the third number for each set. The third number in each triplet should be the largest, representing the potential hypotenuse.

Checking Option 1: 2, 3, 7

Here, $a=2$, $b=3$, and $c=7$. We check if $a^2 + b^2 = c^2$:

$a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13$

$c^2 = 7^2 = 49$

Since $13 \neq 49$, the set (2, 3, 7) is not a Pythagorean triplet.

Checking Option 2: 7, 9, 11

Here, $a=7$, $b=9$, and $c=11$. We check if $a^2 + b^2 = c^2$:

$a^2 + b^2 = 7^2 + 9^2 = 49 + 81 = 130$

$c^2 = 11^2 = 121$

Since $130 \neq 121$, the set (7, 9, 11) is not a Pythagorean triplet.

Checking Option 3: 8, 15, 17

Here, $a=8$, $b=15$, and $c=17$. We check if $a^2 + b^2 = c^2$:

$a^2 + b^2 = 8^2 + 15^2 = 64 + 225 = 289$

$c^2 = 17^2 = 289$

Since $289 = 289$, the set (8, 15, 17) satisfies the Pythagorean theorem. Thus, it is a Pythagorean triplet.

Checking Option 4: 17, 21, 29

Here, $a=17$, $b=21$, and $c=29$. We check if $a^2 + b^2 = c^2$:

$a^2 + b^2 = 17^2 + 21^2 = 289 + 441 = 730$

$c^2 = 29^2 = 841$

Since $730 \neq 841$, the set (17, 21, 29) is not a Pythagorean triplet.

Summary of Checks

Based on the checks, only the set (8, 15, 17) satisfies the condition $a^2 + b^2 = c^2$.

Conclusion

The Pythagorean triplet among the given options is (8, 15, 17).

Revision Table: Key Concepts

Additional Information: Properties of Pythagorean Triplets

Here are some interesting points about Pythagorean triplets:

• There are infinitely many Pythagorean triplets.
• The most famous Pythagorean triplet is (3, 4, 5), because $3^2 + 4^2 = 9 + 16 = 25$, and $5^2 = 25$.
• A Pythagorean triplet $(a, b, c)$ is called a primitive Pythagorean triplet if $a$, $b$, and $c$ are coprime (their greatest common divisor is 1). For example, (3, 4, 5) and (8, 15, 17) are primitive Pythagorean triplets.
• Multiples of primitive Pythagorean triplets are also Pythagorean triplets. For instance, (6, 8, 10) is a Pythagorean triplet because it is $2 \times$ (3, 4, 5). $6^2 + 8^2 = 36 + 64 = 100$, and $10^2 = 100$.
• Euclid's formula can generate Pythagorean triplets. For any two positive integers $m$ and $n$ with $m > n$, the integers $a = m^2 - n^2$, $b = 2mn$, and $c = m^2 + n^2$ form a Pythagorean triplet $(a, b, c)$. If $m$ and $n$ are coprime and one of them is even, the triplet is primitive.