The diagram represents a triangle with height $h$ drawn perpendicular to the base. The problem is to find which of the provided options is equal to $h$, based on the triangle's dimensions.

Given Information:

• $AB = c$, $BC = a$, $AC = b$
• The triangle is divided into two segments by the height $h$:
  • Segment $BD = x$, Segment $DC = y$
  • $x + y = a$

Key Relationships:

1. From the right triangle $\triangle ADB$, we have: $h^2 = b^2 - x^2 \quad \Rightarrow \quad h = \sqrt{b^2 - x^2}$
2. From $\triangle ADC$, we have: $h^2 = c^2 - y^2 \quad \Rightarrow \quad h = \sqrt{c^2 - y^2}$
3. Since $x = \frac{a \cdot c}{b+c}$ and ( y = \frac{X