The lenths of the legs of a right triangle are 3x and `x +1.` The hypotenuse is `3x +1.` What is the value of x ?

Answer

To solve the problem, we will use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

Given:
- Length of one leg (a) = 3x
- Length of the other leg (b) = x + 1
- Length of the hypotenuse (c) = 3x + 1

### Step 1: Apply the Pythagorean theorem
According to the theorem:
\[ a^2 + b^2 = c^2 \]
Substituting the values of a, b, and c:
\[ (3x)^2 + (x + 1)^2 = (3x + 1)^2 \]

### Step 2: Expand the squares
Calculating each term:
\[ (3x)^2 = 9x^2 \]
\[ (x + 1)^2 = x^2 + 2x + 1 \]
\[ (3x + 1)^2 = 9x^2 + 6x + 1 \]

Now substituting these back into the equation:
\[ 9x^2 + (x^2 + 2x + 1) = 9x^2 + 6x + 1 \]

### Step 3: Combine like terms
Combining the left side:
\[ 9x^2 + x^2 + 2x + 1 = 10x^2 + 2x + 1 \]

Now we have:
\[ 10x^2 + 2x + 1 = 9x^2 + 6x + 1 \]

### Step 4: Rearrange the equation
Subtract \(9x^2 + 6x + 1\) from both sides:
\[ 10x^2 + 2x + 1 - 9x^2 - 6x - 1 = 0 \]
This simplifies to:
\[ x^2 - 4x = 0 \]

### Step 5: Factor the equation
Factoring out x:
\[ x(x - 4) = 0 \]

### Step 6: Solve for x
Setting each factor to zero gives us:
1. \(x = 0\)
2. \(x - 4 = 0 \Rightarrow x = 4\)

### Step 7: Determine valid solutions
Since \(x = 0\) would result in one leg of the triangle being \(0\) (which is not possible), we discard this solution. Thus, the only valid solution is:
\[ x = 4 \]

### Final Answer
The value of \(x\) is \(4\).

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