Write a Pythagorean triplet whose two members are 16 & 63

Question

Accepted Answer

To determine the correct Pythagorean triplet involving the numbers 16 and 63, we need to find a third number that satisfies the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this is expressed as:

$$ a^2 + b^2 = c^2 $$

Given the numbers 16 and 63, we need to find a third number (c) such that:

$$ 16^2 + 63^2 = c^2 $$

Calculating the squares of 16 and 63:

$$ 16^2 = 256 $$
$$ 63^2 = 3969 $$

Adding these two results:

$$ 256 + 3969 = 4225 $$

Now, we need to find the square root of 4225 to determine the value of c:

$$ \sqrt{4225} = 65 $$

Therefore, the third number that completes the Pythagorean triplet with 16 and 63 is 65. This confirms that Option 3 is the correct answer.

Now, let's explain why the other options are incorrect:

1. **Option 1: 79**
   - If 79 were the third number, then according to the Pythagorean theorem, we would have:
     $$ 16^2 + 63^2 = 79^2 $$
     Calculating 79 squared:
     $$ 79^2 = 6241 $$
     Since 4225 (the sum of 16 squared and 63 squared) is not equal to 6241, 79 cannot be the correct third number.

2. **Option 2: 32**
   - If 32 were the third number, then:
     $$ 16^2 + 63^2 = 32^2 $$
     Calculating 32 squared:
     $$ 32^2 = 1024 $$
     Since 4225 is not equal to 1024, 32 cannot be the correct third number.

4. **Option 4: 126**
   - If 126 were the third number, then:
     $$ 16^2 + 63^2 = 126^2 $$
     Calculating 126 squared:
     $$ 126^2 = 15876 $$
     Since 4225 is not equal to 15876, 126 cannot be the correct third number.

In conclusion, the only number that satisfies the Pythagorean theorem with 16 and 63 is 65, making Option 3 the correct answer.