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

Answer

To find the hypotenuse of a right triangle with legs in the ratio of 3:4 and an area of 1014 cm², we can follow these steps:

### Step 1: Set up the legs of the triangle
Let the lengths of the legs be represented as:
- One leg = \(3x\)
- Other leg = \(4x\)

### Step 2: Use the area formula
The area \(A\) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For our triangle:
\[ A = \frac{1}{2} \times (3x) \times (4x) \]
This simplifies to:
\[ A = \frac{1}{2} \times 12x^2 = 6x^2 \]

### Step 3: Set the area equal to the given area
We know the area is 1014 cm², so we set up the equation:
\[ 6x^2 = 1014 \]

### Step 4: Solve for \(x^2\)
To find \(x^2\), we divide both sides by 6:
\[ x^2 = \frac{1014}{6} = 169 \]

### Step 5: Solve for \(x\)
Now, take the square root of both sides:
\[ x = \sqrt{169} = 13 \]

### Step 6: Find the lengths of the legs
Now we can find the lengths of the legs:
- One leg = \(3x = 3 \times 13 = 39 \, \text{cm}\)
- Other leg = \(4x = 4 \times 13 = 52 \, \text{cm}\)

### Step 7: Use the Pythagorean theorem to find the hypotenuse
The Pythagorean theorem states:
\[ c^2 = a^2 + b^2 \]
where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs of the triangle. Substituting the values:
\[ c^2 = (39)^2 + (52)^2 \]

### Step 8: Calculate \(39^2\) and \(52^2\)
Calculating the squares:
\[ 39^2 = 1521 \]
\[ 52^2 = 2704 \]

### Step 9: Add the squares
Now, add these two results:
\[ c^2 = 1521 + 2704 = 4225 \]

### Step 10: Find the hypotenuse \(c\)
Taking the square root gives:
\[ c = \sqrt{4225} = 65 \, \text{cm} \]

### Final Answer
The hypotenuse of the triangle is \(65 \, \text{cm}\).

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