In a `Delta ABC, AB=AC=17cm D` is a point on BC such that `AD=15, CD=4 cm`, find BD=?

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Draw Triangle ABC**:
- We have an isosceles triangle ABC where AB = AC = 17 cm.
- Point D is on line segment BC such that AD = 15 cm and CD = 4 cm.

2. **Identify the Lengths**:
- Let BD = x cm.
- Since D is on BC, we can express BC as:
\[ BC = BD + CD = x + 4 \text{ cm} \]

3. **Construct a Perpendicular**:
- Draw a perpendicular line from point A to line segment BC, meeting BC at point P.
- Since triangle ABC is isosceles, this perpendicular also acts as a median, which means BP = PC.

4. **Express Lengths**:
- Let BP = PC = y cm.
- Therefore, we can express:
\[ BD = BP + PD = y + x \text{ cm} \]
\[ CD = PC = y + 4 \text{ cm} \]

5. **Use Pythagorean Theorem**:
- In triangle APD:
\[ AD^2 = AP^2 + PD^2 \implies 15^2 = y^2 + x^2 \implies 225 = y^2 + x^2 \quad \text{(1)} \]
- In triangle APC:
\[ AC^2 = AP^2 + PC^2 \implies 17^2 = y^2 + (x + 4)^2 \implies 289 = y^2 + (x^2 + 8x + 16) \quad \text{(2)} \]

6. **Expand and Simplify**:
- From equation (2):
\[ 289 = y^2 + x^2 + 8x + 16 \]
- Substitute \(y^2 + x^2\) from equation (1) into this:
\[ 289 = 225 + 8x + 16 \]
- Simplifying gives:
\[ 289 = 241 + 8x \implies 8x = 289 - 241 = 48 \implies x = \frac{48}{8} = 6 \text{ cm} \]

7. **Calculate BD**:
- Now, we can find BD:
\[ BD = x + CD = 6 + 4 = 10 \text{ cm} \]

### Final Answer:
BD = 10 cm.