1. **State the problem:** We are given a circle with points A, B, D on a vertical line and a radius labeled $a$. Segments $AD=12$ and $AB=10$ are given. We want to find the equation that results from applying the secant and tangent segment theorem. 2. **Recall the secant and tangent segment theorem:** If a tangent from a point outside the circle touches the circle at one point and a secant from the same point intersects the circle at two points, then: $$\text{(tangent segment)}^2 = \text{(external part of secant)} \times \text{(whole secant)}$$ 3. **Identify segments:** - Tangent segment length = $10$ - Secant segment parts: external part $a$, whole secant $a + 12$ 4. **Apply the theorem:** $$10^2 = a(a + 12)$$ 5. **Rewrite the equation:** $$10^2 = a(a + 12) \implies 100 = a^2 + 12a$$ 6. **Conclusion:** The equation that results from applying the secant and tangent segment theorem is: $$10^2 = a(a + 12)$$ **Final answer:** $$10^2 = a(a + 12)$$