The sides of a triangle `triangleABC` are 15 cm, 41 cm and 52 cm. Find the diameter of the circumcircle of the `triangleABC`

Options

A. 205/3 cm
B. 208/3 cm
C. 69 cm
D. 70 cm

Answer

To find the diameter of the circumcircle of triangle ABC with sides 15 cm, 41 cm, and 52 cm, we will follow these steps:

### Step 1: Calculate the semi-perimeter (s) of the triangle.
The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a = 15 \), \( b = 41 \), and \( c = 52 \).

Calculating \( s \):
\[ s = \frac{15 + 41 + 52}{2} = \frac{108}{2} = 54 \text{ cm} \]

### Step 2: Calculate the area (A) of the triangle using Heron's formula.
Heron's formula states:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{54(54-15)(54-41)(54-52)} \]
Calculating each term:
- \( s - a = 54 - 15 = 39 \)
- \( s - b = 54 - 41 = 13 \)
- \( s - c = 54 - 52 = 2 \)

Now substituting these values back into the area formula:
\[ A = \sqrt{54 \times 39 \times 13 \times 2} \]

### Step 3: Simplify the area calculation.
Calculating the product:
\[ 54 \times 39 = 2106 \]
\[ 2106 \times 13 = 27378 \]
\[ 27378 \times 2 = 54756 \]
Now taking the square root:
\[ A = \sqrt{54756} \]
Calculating the square root gives:
\[ A \approx 234 \text{ cm}^2 \]

### Step 4: Calculate the circumradius (R) using the formula.
The circumradius \( R \) of a triangle can be calculated using the formula:
\[ R = \frac{abc}{4A} \]
Substituting the values:
\[ R = \frac{15 \times 41 \times 52}{4 \times 234} \]

Calculating the numerator:
\[ 15 \times 41 = 615 \]
\[ 615 \times 52 = 31980 \]
Now calculating the denominator:
\[ 4 \times 234 = 936 \]
Now substituting these values back into the formula for \( R \):
\[ R = \frac{31980}{936} \approx 34.2 \text{ cm} \]

### Step 5: Calculate the diameter (D) of the circumcircle.
The diameter \( D \) is given by:
\[ D = 2R \]
Substituting the value of \( R \):
\[ D = 2 \times 34.2 \approx 68.4 \text{ cm} \]

### Final Answer:
The diameter of the circumcircle of triangle ABC is approximately **68.4 cm**.

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