The legs of a right triangle are in the ratio 3 : 4 and its area is `1014cm^(2)` . Find the lengths of its legs.

Answer

To find the lengths of the legs of a right triangle given that the legs are in the ratio 3:4 and the area is 1014 cm², we can follow these steps:

### Step 1: Define the legs in terms of a variable
Let the lengths of the legs of the triangle be represented as:
- One leg = 3x
- Other leg = 4x

### Step 2: Write the formula for the area of a triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In our case, we can take the legs of the triangle as the base and height. Thus, we have:
\[ A = \frac{1}{2} \times (3x) \times (4x) \]

### Step 3: Substitute the area into the equation
We know the area is 1014 cm², so we can set up the equation:
\[ 1014 = \frac{1}{2} \times (3x) \times (4x) \]

### Step 4: Simplify the equation
Now, simplify the right side:
\[ 1014 = \frac{1}{2} \times 12x^2 \]
\[ 1014 = 6x^2 \]

### Step 5: Solve for \( x^2 \)
To isolate \( x^2 \), divide both sides by 6:
\[ x^2 = \frac{1014}{6} \]
\[ x^2 = 169 \]

### Step 6: Solve for \( x \)
Now, take the square root of both sides to find \( x \):
\[ x = \sqrt{169} = 13 \]

### Step 7: Find the lengths of the legs
Now that we have \( x \), we can find the lengths of the legs:
- Length of the first leg = \( 3x = 3 \times 13 = 39 \) cm
- Length of the second leg = \( 4x = 4 \times 13 = 52 \) cm

### Final Answer
The lengths of the legs of the triangle are:
- First leg = 39 cm
- Second leg = 52 cm

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