In `DeltaABC,` AB=10, AC=7, BC=9, then find the length of the median drawn from point Cto side AB.

Answer

To find the length of the median drawn from point C to side AB in triangle ABC, where AB = 10, AC = 7, and BC = 9, we will use the Apollonius theorem. Here are the step-by-step calculations:

### Step 1: Identify the lengths of the sides
We have:
- AB = 10
- AC = 7
- BC = 9

### Step 2: Find the length of half of side AB
Since D is the midpoint of side AB, we can find the length of AD (or DB):
\[ AD = DB = \frac{AB}{2} = \frac{10}{2} = 5 \]

### Step 3: Apply the Apollonius theorem
According to the Apollonius theorem:
\[ AC^2 + BC^2 = 2 \cdot CD^2 + 2 \cdot AD^2 \]
Where:
- \(CD\) is the length of the median we want to find,
- \(AD\) is half of \(AB\).

### Step 4: Substitute the known values into the theorem
Substituting the known values:
\[ 7^2 + 9^2 = 2 \cdot CD^2 + 2 \cdot 5^2 \]
Calculating the squares:
\[ 49 + 81 = 2 \cdot CD^2 + 2 \cdot 25 \]
\[ 130 = 2 \cdot CD^2 + 50 \]

### Step 5: Rearranging the equation
Now, we can rearrange the equation to isolate \(CD^2\):
\[ 130 - 50 = 2 \cdot CD^2 \]
\[ 80 = 2 \cdot CD^2 \]

### Step 6: Solve for \(CD^2\)
Dividing both sides by 2:
\[ CD^2 = \frac{80}{2} = 40 \]

### Step 7: Find \(CD\)
Taking the square root of both sides to find \(CD\):
\[ CD = \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} \]

### Final Answer
The length of the median drawn from point C to side AB is:
\[ CD = 2\sqrt{10} \text{ units} \]

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