Solution to Activity 2: Verifying the Secant-Tangent Theorem

Step 1: Understand the secant-tangent theorem

The secant-tangent theorem states that: $\text{(Length of Tangent Segment)}^2 = \text{(External Secant Segment) * (Total Secant Length)}$

Where:

• $PA$ is the tangent length,
• $PB$ is the external segment of the secant,
• $PC$ is the total length of the secant.

Step 2: Input the given values

From the question:

• $PA = 15$ meters (tangent length),
• $PB = 10$ meters (external secant segment),
• $PC = 25$ meters (total secant length).

We aim to verify if: $PA^2 = PB \cdot PC$

Step 3: Perform calculations

1. Calculate $PA^2$: $PA^2 = 15^2 = 225$

2. Calculate $PB \cdot PC$: $PB \cdot PC = 10 \cdot 25 = 250$

Step 4: Compare the results

• $PA^2 = 225$
• $PB \cdot PC = 250$

Clearly, $PA^2 \neq PB \cdot PC$, meaning the theorem does not hold true for the current design.

Step 5: Explanation and suggestions

The design violates the secant-tangent theorem because the tangent length and secant dimensions do not satisfy the relationship required by the theorem.

Adjustment Suggestion:

• Either increase the tangent length $PA$ to approximately $\sqrt{250} \approx 15.81$ meters, or
• Adjust the secant length $PC$ and/or external segment $PB$ to satisfy $PA^2 = PB \cdot PC$.

---

Questions for Further Exploration:

1. What are the implications of the secant-tangent theorem in real-world applications like architecture?
2. How would increasing the tangent length impact the fountain's design?
3. Can the secant length be reduced instead of increasing the tangent? What would that change geometrically?
4. Are there situations where the secant-tangent theorem doesn’t need to be satisfied? Why or why not?
5. How would the design change if both walkways were tangents?

Tip: Always recheck calculations and verify that lengths are scaled correctly in real-world designs involving geometric theorems.