This image contains a problem about finding the length of unknown segments in the given figures, involving two circles and intersecting line segments. Let us solve each part.

Part 1 (Left Circle - Blue):

Here, we apply the Power of a Point Theorem: $\text{Power of a Point: } (XY) \cdot (XN) = (XK)^2$ Given:

• $XY = X = 6$
• $XN = 2$
• $XK^2 = 5$

Solving ... Power of Point computation for the left circle

Given values

XY = 6 # length of XY XN = 2 # length of XN area = Total valuesApologies for the confusion! I will correct and rephrase. Let’s manually compute the problem.

Solution for Left Circle:

Using the Power of a Point Theorem: $XY \cdot XN = XK^2$ Substitute the given values: $6 \cdot 2 = XK^2$ $12 = XK^2$ $XK = \sqrt{12} = 2\sqrt{3}$

Solution for Right Circle:

Here, the segments are related by: $(SC) \cdot (SB) = (SU)^2$ Substitute the given values:

• $SC = 4x$
• $SB = 15$
• $SU = x$

The equation becomes: $(4x) \cdot (15) = (x)^2$ $60x = x^2$ Simplify: $x^2 - 60x = 0$ $x(x - 60) = 0$ So: $x = 0 \quad \text{or} \quad x = 60$ Since $x > 0$, we take: $x = 60$

Final Results:

1. $XK = 2\sqrt{3}$
2. $x = 60$

Would you like a detailed step-by-step explanation for any of these solutions? 😊

5 Related Questions:

1. What is the Power of a Point Theorem, and how is it applied in geometry?
2. How do we simplify radicals such as $\sqrt{12}$ in algebraic solutions?
3. Can you solve for other unknowns in similar geometric configurations?
4. What are practical uses of intersecting chords and secant lengths in real life?
5. How does the quadratic equation arise in solving geometric problems?

Tip:

Always double-check units and ensure all segments are correctly labeled to avoid confusion during calculations!