Three numbers a,b, and c form a Phythagorean triplet. If a=29, b=21, find c if c is the shortest of the three numbers.

Answer

To find the value of \( c \) in the Pythagorean triplet where \( a = 29 \) and \( b = 21 \), and \( c \) is the shortest side, we can follow these steps:

### Step 1: Identify the sides
In a Pythagorean triplet, the relationship is given by the Pythagorean theorem:
\[ a^2 = b^2 + c^2 \]
Here, \( a \) is the hypotenuse (the longest side), and \( b \) and \( c \) are the other two sides. Since \( a = 29 \) (the longest side) and \( b = 21 \), we need to find \( c \).

### Step 2: Write the equation
Using the Pythagorean theorem:
\[ 29^2 = 21^2 + c^2 \]

### Step 3: Calculate \( 29^2 \) and \( 21^2 \)
First, calculate \( 29^2 \):
\[ 29^2 = 841 \]
Next, calculate \( 21^2 \):
\[ 21^2 = 441 \]

### Step 4: Substitute the values into the equation
Now substitute these values into the equation:
\[ 841 = 441 + c^2 \]

### Step 5: Solve for \( c^2 \)
To find \( c^2 \), rearrange the equation:
\[ c^2 = 841 - 441 \]
Calculate the right side:
\[ c^2 = 400 \]

### Step 6: Find \( c \)
To find \( c \), take the square root of both sides:
\[ c = \sqrt{400} \]
Calculating the square root gives:
\[ c = 20 \]

### Conclusion
Thus, the value of \( c \) is \( 20 \).

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