Point Y lies on line segment XZ. Semicircles are constructed with diameters XY, XZ, and YZ. Find the area of the blue region.

Asymptote code below

[asy]
unitsize(1 cm);

pair A, B, C;

A = (0,0);
B = (2,0);
C = (6,0);

fill(arc((A + C)/2, abs(A - C)/2, 0, 180)--cycle,paleblue);
fill(arc((A + B)/2, abs(A - B)/2, 0, 180)--cycle,white);
fill(arc((B + C)/2, abs(B - C)/2, 0, 180)--cycle,white);
draw(A--B--C--cycle);
draw(arc((A + B)/2, abs(A - B)/2, 0, 180));
draw(arc((A + C)/2, abs(A - C)/2, 0, 180));
draw(arc((B + C)/2, abs(B - C)/2, 0, 180));

label("$X$", A, S);
label("$Y$", B, S);
label("$Z$", C, S);
label("$6$", (A + B)/2, S, red);
label("$10$", (B + C)/2, S, red);
[/asy]

We are tasked with finding the area of the blue region formed by three semicircles constructed on the line segment $XZ$, where point $Y$ lies between $X$ and $Z$. The diameters of the semicircles are $XY$, $YZ$, and $XZ$. Below is the step-by-step approach to solve this problem.

### Step 1: Analyze the Given Diagram
- The points $X$, $Y$, and $Z$ are collinear.
- The semicircles are constructed with diameters $XY$, $YZ$, and $XZ$.
- We are given that the length of segment $XY$ is 2 units, and the length of segment $YZ$ is 4 units (since $XZ = XY + YZ$, and $XZ = 6 + 10 = 16$).

### Step 2: Formula for the Area of a Semicircle
The area of a semicircle is given by the formula:
\[
A_{\text{semicircle}} = \frac{1}{2} \pi r^2
\]
where $r$ is the radius, which is half the diameter.

### Step 3: Areas of the Individual Semicircles
- **Semicircle with diameter $XZ$ (blue region):**
  - The diameter is $XZ = 16$ units, so the radius is $r = 8$ units.
  - The area of this semicircle is:
    \[
    A_{XZ} = \frac{1}{2} \pi (8)^2 = \frac{1}{2} \pi \times 64 = 32\pi
    \]

- **Semicircle with diameter $XY$ (white region):**
  - The diameter is $XY = 6$ units, so the radius is $r = 3$ units.
  - The area of this semicircle is:
    \[
    A_{XY} = \frac{1}{2} \pi (3)^2 = \frac{1}{2} \pi \times 9 = \frac{9\pi}{2}
    \]

- **Semicircle with diameter $YZ$ (white region):**
  - The diameter is $YZ = 10$ units, so the radius is $r = 5$ units.
  - The area of this semicircle is:
    \[
    A_{YZ} = \frac{1}{2} \pi (5)^2 = \frac{1}{2} \pi \times 25 = \frac{25\pi}{2}
    \]

### Step 4: Calculate the Area of the Blue Region
The blue region is the area of the largest semicircle (diameter $XZ$) minus the areas of the two smaller semicircles (diameters $XY$ and $YZ$).

\[
A_{\text{blue}} = A_{XZ} - A_{XY} - A_{YZ}
\]

Substitute the areas:
\[
A_{\text{blue}} = 32\pi - \frac{9\pi}{2} - \frac{25\pi}{2}
\]
First, combine the two smaller areas:
\[
\frac{9\pi}{2} + \frac{25\pi}{2} = \frac{34\pi}{2} = 17\pi
\]
Now subtract this from $32\pi$:
\[
A_{\text{blue}} = 32\pi - 17\pi = 15\pi
\]

### Final Answer:
The area of the blue region is $15\pi$ square units.

---

Do you have any questions, or would you like further details on any of the steps? Here are five follow-up questions to explore the topic further:

1. How would the areas change if the lengths of $XY$ and $YZ$ were different?
2. Can you generalize a formula for the blue area based on different diameters?
3. What if full circles were used instead of semicircles—how would the area calculations change?
4. How does this problem change if point $Y$ is not in the middle of $XZ$?
5. What if the semicircles were on the same side of the line segment instead of alternating?

**Tip:** Always check the relationship between the diameters and radii when dealing with semicircles to ensure accurate area calculations.

Point Y lies on line segment XZ. Semicircles are constructed with diameters XY, XZ, and YZ. Find the area of the blue region.

Solve a geometry problem involving semicircles with diameters XY, YZ, and XZ. Calculate the area of the blue region by applying formulas for the area of semicircles and subtracting the areas of smaller semicircles from the largest. Suitable for students in Grades 8-10.

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