Ratio between two perpendicular side of right angle triangle is 8:15, and it's hypotenuse is 102 m. Find area of right angle triangle?

Options

A. `2460 m ^(2)`
B. `2160m^(2)`
C. `1690 m^(2)`
D. None of these

Answer

To solve the problem step by step, we will follow the given information and apply the necessary mathematical concepts.

### Step 1: Understand the Given Information
We are given:
- The ratio of the two perpendicular sides of a right-angled triangle is 8:15.
- The hypotenuse of the triangle is 102 m.

### Step 2: Assign Variables to the Sides
Let the two perpendicular sides be:
- Side A (shorter side) = 8x
- Side B (longer side) = 15x

### Step 3: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Side A}^2 + \text{Side B}^2 \]
Substituting the values we have:
\[ 102^2 = (8x)^2 + (15x)^2 \]

### Step 4: Calculate the Squares
Calculating the squares:
\[ 102^2 = 10404 \]
\[ (8x)^2 = 64x^2 \]
\[ (15x)^2 = 225x^2 \]

### Step 5: Set Up the Equation
Now, we can set up the equation:
\[ 10404 = 64x^2 + 225x^2 \]
Combine the terms:
\[ 10404 = 289x^2 \]

### Step 6: Solve for x
Now, solve for \( x^2 \):
\[ x^2 = \frac{10404}{289} \]
Calculating this gives:
\[ x^2 = 36 \]
Taking the square root:
\[ x = 6 \]

### Step 7: Find the Lengths of the Sides
Now, substitute \( x \) back to find the lengths of the sides:
- Side A = \( 8x = 8 \times 6 = 48 \) m
- Side B = \( 15x = 15 \times 6 = 90 \) m

### Step 8: Calculate the Area of the Triangle
The area \( A \) of a right-angled triangle is given by:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Here, Base = 90 m and Height = 48 m:
\[ A = \frac{1}{2} \times 90 \times 48 \]
Calculating this gives:
\[ A = \frac{1}{2} \times 4320 = 2160 \text{ m}^2 \]

### Final Answer
The area of the right-angled triangle is **2160 m²**.

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