An isosceles right triangle has area `112.5 m^(2)`. The length of its hypotenuse ( in cm ) is :

Options

A. 21. 213
B. 21.013
C. 21.113
D. 21.313

Answer

To find the length of the hypotenuse of an isosceles right triangle with an area of 112.5 m², we can follow these steps:

### Step 1: Understand the properties of the triangle
In an isosceles right triangle, the two legs (the equal sides) are of the same length, and the area can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Since the base and height are equal in this case, we can denote the length of each leg as \( a \).

### Step 2: Set up the equation for the area
Given that the area of the triangle is 112.5 m², we can write:
\[ 112.5 = \frac{1}{2} \times a \times a \]
This simplifies to:
\[ 112.5 = \frac{1}{2} a^2 \]

### Step 3: Solve for \( a^2 \)
To isolate \( a^2 \), multiply both sides by 2:
\[ 225 = a^2 \]

### Step 4: Find \( a \)
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{225} = 15 \text{ m} \]

### Step 5: Convert meters to centimeters
Since we need the hypotenuse in centimeters, we convert meters to centimeters:
\[ 15 \text{ m} = 15 \times 100 = 1500 \text{ cm} \]

### Step 6: Use the Pythagorean theorem to find the hypotenuse
In an isosceles right triangle, the hypotenuse \( c \) can be calculated using the Pythagorean theorem:
\[ c = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]
Substituting \( a = 1500 \text{ cm} \):
\[ c = 1500 \sqrt{2} \]

### Step 7: Calculate \( c \)
Now, we need to calculate \( 1500 \sqrt{2} \). The approximate value of \( \sqrt{2} \) is 1.414:
\[ c \approx 1500 \times 1.414 \approx 2121 \text{ cm} \]

### Final Answer
Thus, the length of the hypotenuse is approximately:
\[ \text{Hypotenuse} \approx 2121 \text{ cm} \]

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