In an isosceles triangle ABC base AB=6 cm and each lateral side is 5 cm. find the circumradius of the triangle ABC.

Options

A. A)`22/7 cm`
B. B)`25/8 cm`
C. C)`21/8 cm`
D. D)`23/7 cm`

Answer

To find the circumradius \( R \) of the isosceles triangle ABC with base \( AB = 6 \) cm and lateral sides \( AC = BC = 5 \) cm, we can follow these steps:

### Step 1: Identify the triangle and its properties
We have an isosceles triangle ABC where:
- Base \( AB = 6 \) cm
- Lateral sides \( AC = BC = 5 \) cm

### Step 2: Draw the altitude
Draw the altitude \( CD \) from vertex \( C \) to the midpoint \( D \) of base \( AB \). Since \( AB = 6 \) cm, the lengths \( AD \) and \( DB \) will each be \( 3 \) cm.

### Step 3: Apply the Pythagorean theorem
In triangle \( ACD \):
- \( AC^2 = AD^2 + CD^2 \)
- Substitute the known values: \( 5^2 = 3^2 + CD^2 \)
- This simplifies to: \( 25 = 9 + CD^2 \)
- Therefore, \( CD^2 = 25 - 9 = 16 \)
- Thus, \( CD = \sqrt{16} = 4 \) cm.

### Step 4: Calculate the circumradius \( R \)
The formula for the circumradius \( R \) of a triangle is given by:
\[ R = \frac{abc}{4K} \]
where \( a, b, c \) are the lengths of the sides of the triangle and \( K \) is the area of the triangle.

- Here, \( a = 5 \) cm, \( b = 5 \) cm, \( c = 6 \) cm.
- To find the area \( K \), we can use the formula for the area of a triangle:
\[ K = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2. \]

### Step 5: Substitute values into the circumradius formula
Now substituting the values into the circumradius formula:
\[ R = \frac{5 \times 5 \times 6}{4 \times 12} = \frac{150}{48} = \frac{25}{8} \text{ cm}. \]

### Final Answer
Thus, the circumradius \( R \) of triangle ABC is \( \frac{25}{8} \) cm or \( 3.125 \) cm.

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