which set of numbers represent a Pythagorean triplets
6,
To determine if a set of numbers represents a Pythagorean triplet, you need to check if the three numbers satisfy the Pythagorean theorem. According to the theorem, in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
So, let's check if the numbers 6, ?, ? form a Pythagorean triplet.
1. First, consider the number 6 as the length of one side (a) of the triangle.
2. Next, we need to find the other two sides (b and c).
3. According to the Pythagorean theorem, a^2 + b^2 = c^2, where c is the length of the hypotenuse.
4. Since we don't have the lengths of b and c, let's consider them as variables: b and c.
5. Now, we can write the equation as 6^2 + b^2 = c^2.
Simplifying further, 36 + b^2 = c^2.
6. To check if this equation represents a Pythagorean triplet, we need to find possible values for b and c that satisfy the equation.
One way to find Pythagorean triplets is to use trial and error, but there are also some known patterns:
- A common Pythagorean triplet is the (3, 4, 5) triplet, where 3^2 + 4^2 = 5^2. These numbers form a Pythagorean triplet because they satisfy the equation.
- Another common triplet is the (5, 12, 13) triplet, where 5^2 + 12^2 = 13^2. This triplet also follows the Pythagorean theorem.
Now, let's substitute some values for b and c using the known Pythagorean triplets:
1. Substitute b = 3: 36 + 3^2 = 45.
The equation becomes 36 + 9 = 45.
This is not a valid Pythagorean triplet.
2. Substitute b = 4: 36 + 4^2 = 52.
The equation becomes 36 + 16 = 52.
This is not a valid Pythagorean triplet either.
3. Substitute b = 5: 36 + 5^2 = 61.
The equation becomes 36 + 25 = 61.
This is also not a valid Pythagorean triplet.
Therefore, the set of numbers {6, ?, ?} does not represent a Pythagorean triplet, as none of the possible solutions satisfy the Pythagorean theorem.