The longest side of a right angled triangle is 4cm longer than one side and 2 cm longer than the other side. Find the longest side.

Answer

To solve the problem step-by-step, we will define the variables and use the properties of a right-angled triangle.

### Step 1: Define the Variables
Let the longest side (hypotenuse) of the right-angled triangle be denoted as \( x \) cm. According to the problem:
- One side (let's call it \( a \)) is \( x - 4 \) cm (4 cm shorter than the longest side).
- The other side (let's call it \( b \)) is \( x - 2 \) cm (2 cm shorter than the longest side).

### Step 2: Apply the Pythagorean Theorem
In a right-angled triangle, the relationship between the lengths of the sides is given by the Pythagorean theorem:
\[ x^2 = a^2 + b^2 \]
Substituting the expressions for \( a \) and \( b \):
\[ x^2 = (x - 4)^2 + (x - 2)^2 \]

### Step 3: Expand the Squares
Now, we will expand the squares on the right-hand side:
\[ x^2 = (x - 4)^2 + (x - 2)^2 \]
\[ = (x^2 - 8x + 16) + (x^2 - 4x + 4) \]
Combining like terms:
\[ x^2 = x^2 - 8x + 16 + x^2 - 4x + 4 \]
\[ = 2x^2 - 12x + 20 \]

### Step 4: Rearrange the Equation
Now, we will rearrange the equation to set it to zero:
\[ x^2 - (2x^2 - 12x + 20) = 0 \]
\[ 0 = 2x^2 - 12x + 20 - x^2 \]
\[ 0 = x^2 - 12x + 20 \]

### Step 5: Factor the Quadratic Equation
Next, we will factor the quadratic equation:
\[ x^2 - 12x + 20 = 0 \]
We need two numbers that multiply to \( 20 \) and add up to \( -12 \). These numbers are \( -10 \) and \( -2 \):
\[ (x - 10)(x - 2) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x - 10 = 0 \) → \( x = 10 \)
2. \( x - 2 = 0 \) → \( x = 2 \)

### Step 7: Determine the Valid Solution
Since \( x \) represents the length of the longest side, we need to ensure that it is a positive value. If we substitute \( x = 2 \) back into our expressions for the sides:
- \( a = 2 - 4 = -2 \) (not valid, as length cannot be negative)
- \( b = 2 - 2 = 0 \) (not valid, as length cannot be zero)

Thus, the only valid solution is:
\[ x = 10 \text{ cm} \]

### Conclusion
The longest side of the right-angled triangle is \( 10 \) cm.

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