O is the circum centre of the triangle ABC with circumradius 13 cm. Let BC = 24 cm and OD is perpendicular to BC. Then the length of OD is

Options

A. 3 cm
B. 4 cm
C. 5 cm
D. 7 cm

Answer

To find the length of \( OD \), where \( O \) is the circumcenter of triangle \( ABC \), we can use the properties of circles and triangles. Here’s a step-by-step solution:

### Step 1: Identify the Given Information
- Circumradius \( R = 13 \) cm
- Length of side \( BC = 24 \) cm

### Step 2: Understand the Geometry
Since \( O \) is the circumcenter, it is the center of the circumcircle of triangle \( ABC \). The line \( OD \) is perpendicular to \( BC \) and bisects \( BC \) at point \( D \).

### Step 3: Calculate the Length of \( BD \) and \( DC \)
Since \( D \) is the midpoint of \( BC \):
\[ BD = DC = \frac{BC}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 4: Apply the Pythagorean Theorem
In triangle \( ODB \), we can apply the Pythagorean theorem:
\[ OB^2 = OD^2 + BD^2 \]
Where:
- \( OB = R = 13 \) cm
- \( BD = 12 \) cm

Substituting the values:
\[ 13^2 = OD^2 + 12^2 \]
\[ 169 = OD^2 + 144 \]

### Step 5: Solve for \( OD^2 \)
Rearranging the equation to isolate \( OD^2 \):
\[ OD^2 = 169 - 144 \]
\[ OD^2 = 25 \]

### Step 6: Find the Length of \( OD \)
Taking the square root of both sides:
\[ OD = \sqrt{25} = 5 \text{ cm} \]

### Final Answer
The length of \( OD \) is \( 5 \) cm.

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