Let's solve each part step by step!

(a) Two Chords Intersect Inside the Circle

In this problem, two chords, $EK$ and $MJ$, intersect at point $E$ inside the circle. We are given:

• $EK = 16.5$
• $EJ = 22$
• $EN = 33$

We need to find the length of $MN$, using the intersecting chords theorem, which states:

$\text{Product of the lengths of the segments of one chord} = \text{Product of the lengths of the segments of the other chord}$

For chord $EK$ and chord $MJ$, the segments are:

• $EK = EJ + EN = 16.5$
• $MJ = ME + EN = 33$

From the intersection formula:

$(EJ) \times (EN) = (MJ) \times (MN)$

Substitute the known values:

$22 \times 33 = 16.5 \times MN$

Now solve for $MN$:

$726 = 16.5 \times MN$

$MN = \frac{726}{16.5} = 44$

So, $MN = 44$.

(b) Secant and Tangent Problem

In the second part, a tangent and a secant are drawn from point $U$ to the circle, and we are given:

• $UG = 15$ (the length of the tangent)
• $UC = 10$ (the length of the secant)

We are asked to find $UD$, which is the total length from the exterior point $U$ to the farthest point of the secant.

The secant-tangent theorem says:

$\text{Tangent}^2 = \text{Secant} \times \text{External segment of secant}$

From this, we can express the equation for $UD$:

$UG^2 = UC \times UD$

Substitute the known values:

$15^2 = 10 \times UD$

$225 = 10 \times UD$

Solve for $UD$:

$UD = \frac{225}{10} = 22.5$

So, $UD = 22.5$.

Would you like any further clarifications or more steps on similar problems? Let me know if you need further explanation!

Follow-up questions:

1. What is the relationship between the intersecting chords theorem and the power of a point?
2. How can you derive the secant-tangent theorem from the intersecting secant theorem?
3. What happens if two tangents are drawn from the same external point?
4. How can the length of a chord be determined if only its endpoints are known?
5. What other geometrical properties apply when a tangent intersects a circle?

Tip: The intersecting chords theorem and the secant-tangent theorem are powerful tools for solving many circle-related problems, especially in geometry!