In `DeltaABC, AB^(2)+AC^(2)=2500cm^(2)` and median AD = 25 cm, Find BC:

Options

A. 25 cm
B. 40 cm
C. 50 cm
D. 48 cm

Answer

To find the length of side BC in triangle ABC, given that \( AB^2 + AC^2 = 2500 \, \text{cm}^2 \) and the median \( AD = 25 \, \text{cm} \), we can use Apollonius's theorem. Here’s the step-by-step solution:

### Step 1: Understand Apollonius's Theorem
Apollonius's theorem states that in any triangle, the sum of the squares of the two sides is equal to twice the square of the median plus half the square of the third side. Mathematically, it can be expressed as:
\[ AB^2 + AC^2 = 2AD^2 + \frac{1}{2} BC^2 \]

### Step 2: Substitute the Known Values
From the problem, we know:
- \( AB^2 + AC^2 = 2500 \, \text{cm}^2 \)
- \( AD = 25 \, \text{cm} \)

Now, we can substitute these values into the equation:
\[ 2500 = 2(25^2) + \frac{1}{2} BC^2 \]

### Step 3: Calculate \( 2AD^2 \)
Calculate \( 25^2 \):
\[ 25^2 = 625 \]
Now, substitute this back into the equation:
\[ 2500 = 2(625) + \frac{1}{2} BC^2 \]
\[ 2500 = 1250 + \frac{1}{2} BC^2 \]

### Step 4: Isolate \( \frac{1}{2} BC^2 \)
Subtract \( 1250 \) from both sides:
\[ 2500 - 1250 = \frac{1}{2} BC^2 \]
\[ 1250 = \frac{1}{2} BC^2 \]

### Step 5: Solve for \( BC^2 \)
Multiply both sides by \( 2 \):
\[ BC^2 = 1250 \times 2 \]
\[ BC^2 = 2500 \]

### Step 6: Find \( BC \)
Take the square root of both sides:
\[ BC = \sqrt{2500} \]
\[ BC = 50 \, \text{cm} \]

### Final Answer
Thus, the length of side \( BC \) is \( 50 \, \text{cm} \).

---