The perimeter of a certain isosceles right triangle is `10 + 10sqrt(2)cm`. What is the length of the hypotenuse of the triangle ?

Options

A. 5 cm
B. 10 cm
C. `5sqrt(2) cm`
D. `10sqrt(2) cm`

Answer

To find the length of the hypotenuse of the isosceles right triangle with a given perimeter of \(10 + 10\sqrt{2}\) cm, we can follow these steps:

### Step 1: Understand the properties of the isosceles right triangle
In an isosceles right triangle, the two legs are equal in length. Let's denote the length of each leg as \(a\). The hypotenuse can be expressed using the Pythagorean theorem.

### Step 2: Write the expression for the hypotenuse
For an isosceles right triangle:
\[ \text{Hypotenuse} = a\sqrt{2} \]

### Step 3: Write the expression for the perimeter
The perimeter \(P\) of the triangle can be expressed as:
\[ P = a + a + a\sqrt{2} = 2a + a\sqrt{2} \]

### Step 4: Set the perimeter equal to the given value
We know the perimeter is \(10 + 10\sqrt{2}\):
\[ 2a + a\sqrt{2} = 10 + 10\sqrt{2} \]

### Step 5: Factor out \(a\) from the left side
We can factor out \(a\) from the left side:
\[ a(2 + \sqrt{2}) = 10 + 10\sqrt{2} \]

### Step 6: Solve for \(a\)
To isolate \(a\), we divide both sides by \(2 + \sqrt{2}\):
\[ a = \frac{10 + 10\sqrt{2}}{2 + \sqrt{2}} \]

### Step 7: Simplify the expression for \(a\)
To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ a = \frac{(10 + 10\sqrt{2})(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})} \]
Calculating the denominator:
\[ (2 + \sqrt{2})(2 - \sqrt{2}) = 4 - 2 = 2 \]
Now for the numerator:
\[ (10 + 10\sqrt{2})(2 - \sqrt{2}) = 20 - 10\sqrt{2} + 20\sqrt{2} - 20 = 10\sqrt{2} \]
So we have:
\[ a = \frac{10\sqrt{2}}{2} = 5\sqrt{2} \]

### Step 8: Calculate the hypotenuse
Now we can find the hypotenuse using the formula:
\[ \text{Hypotenuse} = a\sqrt{2} = 5\sqrt{2} \cdot \sqrt{2} = 5 \cdot 2 = 10 \text{ cm} \]

### Final Answer
The length of the hypotenuse of the triangle is \(10\) cm.

---