If the area of a triangle is `1176 cm^(2)` and base : corresponding altitude is 3:4, then the altitude of the triangle is: 

Options

A. 42 cm
B. 52 cm
C. 54 cm
D. 56 cm

Answer

To find the altitude of the triangle given the area and the ratio of the base to the altitude, we can follow these steps:

### Step 1: Understand the given information
We know that:
- The area of the triangle is \(1176 \, \text{cm}^2\).
- The ratio of the base to the corresponding altitude is \(3:4\).

### Step 2: Assign variables based on the ratio
Let the base be \(3x\) and the altitude be \(4x\), where \(x\) is a common multiplier.

### Step 3: Use the formula for the area of a triangle
The formula for the area of a triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values we have:
\[ 1176 = \frac{1}{2} \times (3x) \times (4x) \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 1176 = \frac{1}{2} \times 12x^2 \]
\[ 1176 = 6x^2 \]

### Step 5: Solve for \(x^2\)
To isolate \(x^2\), divide both sides by 6:
\[ x^2 = \frac{1176}{6} \]
\[ x^2 = 196 \]

### Step 6: Solve for \(x\)
Now, take the square root of both sides:
\[ x = \sqrt{196} = 14 \]

### Step 7: Find the altitude
Now that we have \(x\), we can find the altitude:
\[ \text{Altitude} = 4x = 4 \times 14 = 56 \, \text{cm} \]

### Final Answer
The altitude of the triangle is \(56 \, \text{cm}\).

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