The problem involves finding the length of the unknown segment ($x$) in two different figures using geometric principles. Let's analyze and solve them:

Problem 5

Given:
A circle where two chords intersect inside the circle.

• One part of the first chord is $4$, and the other part is $x$.
• The two parts of the second chord are $5$ and $10$.

Formula:

The product of the two segments of one chord equals the product of the two segments of the other chord:
$\text{(Part 1 of Chord 1)} \cdot \text{(Part 2 of Chord 1)} = \text{(Part 1 of Chord 2)} \cdot \text{(Part 2 of Chord 2)}$

Calculation:

$4 \cdot x = 5 \cdot 10$ $4x = 50$ $x = \frac{50}{4} = 12.5$

So, $x = 12.5$.

Problem 6

Given:
A circle where a secant and a tangent intersect outside the circle.

• The tangent segment is $4$.
• The external part of the secant is $12$, and the total secant length is $x$.

Formula:

$\text{Tangent}^2 = \text{External part of secant} \cdot \text{Total secant length}$

Calculation:

$4^2 = 12 \cdot x$ $16 = 12x$ $x = \frac{16}{12} = \frac{4}{3} \approx 1.33$

So, $x \approx 1.33$.

Final Answers:

1. For Problem 5: $x = 12.5$.
2. For Problem 6: $x \approx 1.33$.

Would you like a more detailed explanation or further clarifications?

Related Questions:

1. How does the chord-segment product rule apply to other cases?
2. What is the geometric proof of the tangent-secant relationship?
3. How do you generalize these formulas for multiple secants or chords?
4. Can these principles be applied in 3D geometry, such as spheres?
5. What are common errors when using these formulas in exams?

Tip:

Always ensure you correctly identify the lengths of the segments—mixing up the external and total lengths in secant problems is a common mistake!