The hypotenuse of a right triangle is 13 cm and the difference between the other two sides is 7cm. Find the two unknown side of the triangle.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let the lengths of the two unknown sides of the right triangle be \( AB \) and \( BC \). We know that the hypotenuse \( AC = 13 \, \text{cm} \) and the difference between the two sides is \( 7 \, \text{cm} \).

### Step 2: Set up the equations
Let \( BC = x \) (the longer side) and \( AB = x - 7 \) (the shorter side).

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ 13^2 = (x - 7)^2 + x^2 \]

### Step 4: Expand the equation
Calculating \( 13^2 \):
\[ 169 = (x - 7)^2 + x^2 \]
Expanding \( (x - 7)^2 \):
\[ 169 = (x^2 - 14x + 49) + x^2 \]
Combining like terms:
\[ 169 = 2x^2 - 14x + 49 \]

### Step 5: Rearrange the equation
Rearranging the equation to set it to zero:
\[ 2x^2 - 14x + 49 - 169 = 0 \]
This simplifies to:
\[ 2x^2 - 14x - 120 = 0 \]

### Step 6: Simplify the equation
Dividing the entire equation by 2:
\[ x^2 - 7x - 60 = 0 \]

### Step 7: Factor the quadratic equation
We need to factor \( x^2 - 7x - 60 \). We look for two numbers that multiply to \(-60\) and add to \(-7\). These numbers are \(-12\) and \(5\):
\[ (x - 12)(x + 5) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 12 = 0 \quad \text{or} \quad x + 5 = 0 \]
Thus:
\[ x = 12 \quad \text{or} \quad x = -5 \]

### Step 9: Determine the valid solution
Since \( x \) represents a length, we discard \( x = -5 \). Therefore, \( x = 12 \).

### Step 10: Find the lengths of the sides
Now substituting back to find the lengths of the sides:
- \( BC = x = 12 \, \text{cm} \)
- \( AB = x - 7 = 12 - 7 = 5 \, \text{cm} \)

### Final Answer
The lengths of the two unknown sides of the triangle are:
- \( BC = 12 \, \text{cm} \)
- \( AB = 5 \, \text{cm} \)

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