write a Pythagorean triplet whose two members are 16 & 63 ?

Options

A. 79
B. 32
C. 65
D. 126

Answer

To find a Pythagorean triplet where two members are 16 and 63, we can follow these steps:

### Step 1: Understand the Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. This can be expressed as:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

### Step 2: Calculate the squares of the given numbers
We need to calculate the squares of 16 and 63:
- The square of 16:
\[ 16^2 = 256 \]
- The square of 63:
\[ 63^2 = 3969 \]

### Step 3: Determine the possible hypotenuse
We have two scenarios to consider:
1. If 63 is the hypotenuse, then we need to find the missing side \( b \) using the equation:
\[ 63^2 = 16^2 + b^2 \]
2. If 16 is the hypotenuse, then we need to find the missing side \( a \) using the equation:
\[ 16^2 = 63^2 + a^2 \]

### Step 4: Check the first scenario (63 as hypotenuse)
Using the first scenario:
\[ 3969 = 256 + b^2 \]
Subtract 256 from both sides:
\[ b^2 = 3969 - 256 \]
\[ b^2 = 3713 \]
Now, we need to check if 3713 is a perfect square. The square root of 3713 is approximately 60.9, which is not an integer. Therefore, this scenario does not yield a valid triplet.

### Step 5: Check the second scenario (16 as hypotenuse)
Using the second scenario:
\[ 256 = 3969 + a^2 \]
This is not possible since 256 is less than 3969. Therefore, this scenario is also invalid.

### Step 6: Find the missing side using the Pythagorean triplet formula
Since neither scenario worked, we can use the formula for generating Pythagorean triplets. The formula for generating a triplet is:
\[ (m^2 - n^2, 2mn, m^2 + n^2) \]
where \( m \) and \( n \) are integers with \( m > n > 0 \).

### Step 7: Find suitable \( m \) and \( n \)
We need to find integers \( m \) and \( n \) such that one of the generated sides equals 16 or 63.

If we try \( m = 8 \) and \( n = 7 \):
- \( m^2 - n^2 = 8^2 - 7^2 = 64 - 49 = 15 \)
- \( 2mn = 2 \times 8 \times 7 = 112 \)
- \( m^2 + n^2 = 8^2 + 7^2 = 64 + 49 = 113 \)

This does not yield 16 or 63.

### Step 8: Find the correct triplet
After testing various combinations, we find that:
If we take \( m = 8 \) and \( n = 1 \):
- \( m^2 - n^2 = 8^2 - 1^2 = 64 - 1 = 63 \)
- \( 2mn = 2 \times 8 \times 1 = 16 \)
- \( m^2 + n^2 = 8^2 + 1^2 = 64 + 1 = 65 \)

Thus, the Pythagorean triplet is:
\[ (16, 63, 65) \]

### Final Answer
The Pythagorean triplet whose two members are 16 and 63 is \( (16, 63, 65) \).

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