`angle ACB` is an angle in the semicircle of diameter AB = 5 and AC : BC = 3:4. The area of the triangle ABC is

Options

A. `6sqrt(2)` sq. cm
B. 4 sq. cm
C. 12 sq. cm
D. 6 sq. cm

Answer

To find the area of triangle ABC where angle ACB is inscribed in a semicircle with diameter AB = 5 and the ratio of sides AC to BC is 3:4, we can follow these steps:

### Step 1: Understand the Geometry
Since AB is the diameter of the semicircle, angle ACB is a right angle (90 degrees) according to the inscribed angle theorem.

### Step 2: Assign Lengths Based on the Ratio
Given that the ratio of AC to BC is 3:4, we can assign:
- AC = 3x
- BC = 4x

### Step 3: Use the Pythagorean Theorem
In right triangle ACB, we can apply the Pythagorean theorem:
\[ AB^2 = AC^2 + BC^2 \]
Substituting the known values:
\[ 5^2 = (3x)^2 + (4x)^2 \]
This simplifies to:
\[ 25 = 9x^2 + 16x^2 \]
\[ 25 = 25x^2 \]

### Step 4: Solve for x
Dividing both sides by 25:
\[ 1 = x^2 \]
Taking the square root:
\[ x = 1 \]

### Step 5: Find the Lengths of AC and BC
Now substituting x back into our expressions for AC and BC:
- AC = 3x = 3(1) = 3
- BC = 4x = 4(1) = 4

### Step 6: Calculate the Area of Triangle ABC
The area \( A \) of triangle ABC can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Here, we can take AC as the base and BC as the height:
\[ A = \frac{1}{2} \times 4 \times 3 \]
\[ A = \frac{1}{2} \times 12 = 6 \]

Thus, the area of triangle ABC is \( 6 \) square units.

### Final Answer
The area of triangle ABC is \( 6 \, \text{cm}^2 \).

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