5. A rectangle has two opposite points at points (1: 2) and (5; 5). If the other vertices are on the line x = 3, then the coordinates of the vertex near the x-axis are

We are given two opposite points of a rectangle: \( (1, 2) \) and \( (5, 5) \), and we know that the other two vertices lie on the line \( x = 3 \).

### Step 1: Geometry and Conditions
- The two points \( (1, 2) \) and \( (5, 5) \) represent opposite corners of the rectangle.
- The line \( x = 3 \) is a vertical line where the other two vertices of the rectangle must lie.

### Step 2: Properties of a Rectangle
A rectangle has perpendicular sides. Since the two vertices are on the line \( x = 3 \), the sides along this line are vertical, while the other sides are horizontal or sloped. To calculate the remaining points, we need to explore the symmetry and distances involved.

### Step 3: Midpoint and Symmetry
The midpoint of the diagonal (joining \( (1, 2) \) and \( (5, 5) \)) gives the center of the rectangle. The midpoint formula for two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Using this formula:

\[
M = \left( \frac{1 + 5}{2}, \frac{2 + 5}{2} \right) = \left( 3, \frac{7}{2} \right) = \left( 3, 3.5 \right)
\]

Thus, the center of the rectangle is \( (3, 3.5) \).

### Step 4: Finding the Other Vertices
Since the rectangle is symmetric about this midpoint, and the other vertices lie on the line \( x = 3 \), their \( x \)-coordinates will be 3. The \( y \)-coordinates will be symmetric about \( y = 3.5 \). 

- The distance from the midpoint to \( (1, 2) \) or \( (5, 5) \) is the vertical component of their separation. This distance is:

\[
\text{Distance in } y = 3.5 - 2 = 1.5
\]

Thus, the two remaining vertices will be at \( (3, 3.5 + 1.5) = (3, 5) \) and \( (3, 3.5 - 1.5) = (3, 2) \).

### Step 5: Vertex Near the X-axis
The vertex closest to the x-axis has the smallest \( y \)-coordinate. Therefore, the required vertex is:

\[
(3, 2)
\]

### Final Answer:
The coordinates of the vertex near the x-axis are \( \mathbf{(3, 2)} \).

Would you like more details on the steps or have further questions?

Here are 5 related questions to explore:
1. How do you find the midpoint between two points in a plane?
2. What is the importance of symmetry in solving geometric problems like this one?
3. Can you explain how the distance formula applies to finding the distance between two points?
4. How would you find the length of the diagonal of this rectangle?
5. What is the general equation of a line perpendicular to a given line in geometry?

**Tip:** Midpoint and symmetry are very useful when dealing with rectangles, as they can simplify finding unknown vertices or sides.

A rectangle has two opposite points at points (1, 2) and (5, 5). If the other vertices are on the line x = 3, then the coordinates of the vertex near the x-axis are.

Learn how to find the vertex of a rectangle when given two opposite points at (1, 2) and (5, 5), and where the other vertices lie on the line x = 3. This problem involves symmetry, the midpoint formula, and properties of rectangles. Suitable for students in Grades 9-10.

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