The three circle problems you've provided involve the application of the intersecting chords theorem, which states:

$\text{(Segment 1)} \times \text{(Segment 2)} = \text{(Segment 3)} \times \text{(Segment 4)}$

For each problem, two chords intersect inside the circle, and we need to find the unknown value $x$ by applying this formula.

Problem 4:

In this circle, the intersecting chords are $NK$ and $ML$, intersecting at point $P$.

• $NP = 12$
• $PK = x$
• $MP = 12$
• $PL = 6$

Using the intersecting chords theorem: $NP \times PK = MP \times PL$ Substituting the known values: $12 \times x = 12 \times 6$ $12x = 72 \quad \Rightarrow \quad x = \frac{72}{12} = 6$

Problem 5:

In this circle, the intersecting chords are $AD$ and $EC$, intersecting at point $B$.

• $AB = 6$
• $BD = x$
• $EB = 9$
• $BC = 6$

Using the intersecting chords theorem: $AB \times BD = EB \times BC$ Substituting the known values: $6 \times x = 9 \times 6$ $6x = 54 \quad \Rightarrow \quad x = \frac{54}{6} = 9$

Problem 6:

In this circle, the intersecting chords are $RS$ and $UT$, intersecting at point $V$.

• $RV = 6$
• $VS = x$
• $UV = 8$
• $VT = 4$

Using the intersecting chords theorem: $RV \times VS = UV \times VT$ Substituting the known values: $6 \times x = 8 \times 4$ $6x = 32 \quad \Rightarrow \quad x = \frac{32}{6} = \frac{16}{3} \approx 5.33$

Do you have any questions or would you like more details?

Here are 5 related questions:

1. What is the intersecting chords theorem used for?
2. How would the solution change if the lengths of the chords were different?
3. What are other theorems that apply to intersecting lines in circles?
4. How do you derive the intersecting chords theorem?
5. Can this theorem be used if the intersection point is outside the circle?

Tip: Always make sure to cross-check the unit consistency when applying formulas in geometry to avoid simple mistakes.