In a `triangleABC`, a perpendicular OD is drawn on the side AC from the mid-point of the side AB. If BC=`root(4)(3) cm`, find the length of OD.

Options

A. 2 cm
B. 3 cm
C. 4 cm
D. none of these

Answer

To find the length of OD in triangle ABC, we will follow these steps:

### Step 1: Understand the Triangle and Given Information
We have triangle ABC with a perpendicular OD drawn from the midpoint D of side AB to side AC. We are given that the length of side BC is \( \sqrt{4} \cdot 3 \) cm, which simplifies to \( 4\sqrt{3} \) cm.

### Step 2: Identify the Relationship Between the Triangles
Since D is the midpoint of AB, we can denote:
- AD = DB = x (where x is the length of half of AB).
- Therefore, AB = 2x.

### Step 3: Use Similar Triangles
Triangles ABC and AOD are similar because OD is perpendicular to AC. This gives us the following proportion:
\[ \frac{AD}{AB} = \frac{OD}{CD} \]

### Step 4: Substitute the Known Values
From the previous step, we know:
- AD = x
- AB = 2x
- CD = BC = \( 4\sqrt{3} \) cm

Substituting these into the proportion:
\[ \frac{x}{2x} = \frac{OD}{4\sqrt{3}} \]

### Step 5: Simplify the Proportion
The left side simplifies to:
\[ \frac{1}{2} = \frac{OD}{4\sqrt{3}} \]

### Step 6: Cross-Multiply to Solve for OD
Cross-multiplying gives:
\[ OD = \frac{1}{2} \cdot 4\sqrt{3} \]
\[ OD = 2\sqrt{3} \]

### Step 7: Calculate the Approximate Value of OD
To find the approximate numerical value, we can use the approximation \( \sqrt{3} \approx 1.732 \):
\[ OD \approx 2 \cdot 1.732 \approx 3.464 \text{ cm} \]

### Step 8: Round to the Nearest Whole Number
Rounding 3.464 gives us approximately 3 cm.

### Final Answer
Thus, the length of OD is approximately **3 cm**.

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