Sketch the region bounded by the following curves and determine the centroid of the region using the symmetry of the region. y=|x−10| and y=6−|x−10|

We are given two curves, \( y = |x - 10| \) and \( y = 6 - |x - 10| \), and we need to:

1. **Sketch the region bounded by the curves**.
2. **Determine the centroid** using symmetry.

### 1. **Understanding the curves**:
   - The function \( y = |x - 10| \) is a V-shaped function that opens upwards with its vertex at \( (10, 0) \).
   - The function \( y = 6 - |x - 10| \) is a downward-opening V-shaped function with a vertex at \( (10, 6) \).

#### Finding the intersection points:
To find the intersection points of the two curves, we solve for \( y \) when the two expressions are equal:

\[
|x - 10| = 6 - |x - 10|
\]
This simplifies to:

\[
2|x - 10| = 6
\]
\[
|x - 10| = 3
\]

Thus, the solutions are:

\[
x - 10 = 3 \quad \text{or} \quad x - 10 = -3
\]
\[
x = 13 \quad \text{or} \quad x = 7
\]

So, the curves intersect at \( x = 7 \) and \( x = 13 \). The corresponding \( y \)-value for both points is:

\[
y = |7 - 10| = 3 \quad \text{or} \quad y = |13 - 10| = 3
\]

Thus, the intersection points are \( (7, 3) \) and \( (13, 3) \).

### 2. **Sketch of the region**:
- The function \( y = |x - 10| \) forms a V-shape with the vertex at \( (10, 0) \).
- The function \( y = 6 - |x - 10| \) forms an inverted V-shape with the vertex at \( (10, 6) \).
- The region is bounded between these curves from \( x = 7 \) to \( x = 13 \), with the intersection points at \( (7, 3) \) and \( (13, 3) \).

### 3. **Centroid of the region**:
We can use symmetry to determine the centroid:

- **Symmetry in the \( x \)-axis**: The region is symmetric about \( x = 10 \) (the vertical line through the vertices of the curves). Therefore, the \( x \)-coordinate of the centroid is \( x = 10 \).
- **Symmetry in the \( y \)-axis**: The region is symmetric vertically between \( y = 0 \) and \( y = 6 \), so the \( y \)-coordinate of the centroid is halfway between \( y = 0 \) and \( y = 6 \), which is \( y = 3 \).

Thus, the centroid is located at \( (10, 3) \).

### Final Answer:
The centroid of the region bounded by the curves \( y = |x - 10| \) and \( y = 6 - |x - 10| \) is \( (10, 3) \).

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Would you like further details on the region's geometry or steps in determining the centroid? Here are some related questions for further exploration:

1. How do you find the area of the region bounded by the curves?
2. Can you calculate the centroid using integration?
3. What is the significance of symmetry in centroid calculations?
4. How do you find the distance between the vertices of the two curves?
5. How do absolute value functions behave geometrically?

**Tip**: Symmetry in geometry can simplify calculations like centroids and areas significantly!

Sketch the region bounded by the following curves and determine the centroid of the region using the symmetry of the region. y = |x−10| and y = 6−|x−10|

Explore how to sketch the region bounded by the curves y = |x−10| and y = 6−|x−10|, and find the centroid using symmetry. This problem covers key algebraic concepts, including absolute value functions and the use of symmetry in determining centroids. Ideal for students in Grades 10-12.

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