Verify the given geometric property.
Use slopes to show that 
A(3, 2), B(13, 4), C(12, 9),
 and 
D(2, 7)
 are vertices of a rectangle.
We first plot the points to find the pairs of points that determine each side. Next we find the slopes of the four sides. The slope of AB is   , the slope of AD is   , the slope of BC is   , and the slope of CD is   . Two sides are parallel if their slopes are  . Thus, we see that AB and  are parallel to each other and AD and  are parallel to each other. Two lines are perpendicular to one another when the product of their slopes is equal to   . Since (slope of AB) ✕ (slope of AD) =   , the sides AB and AD are perpendicular to each other. Therefore, the points A, B, C, and D are vertices of a rectangle.

To verify that points \( A(3, 2) \), \( B(13, 4) \), \( C(12, 9) \), and \( D(2, 7) \) are the vertices of a rectangle, we will use the following steps:

### Step 1: Calculate the slopes of the sides
The slope of a line between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We will compute the slopes of the following sides: \( AB \), \( AD \), \( BC \), and \( CD \).

1. **Slope of AB** (between \( A(3, 2) \) and \( B(13, 4) \)):
\[
m_{AB} = \frac{4 - 2}{13 - 3} = \frac{2}{10} = \frac{1}{5}
\]

2. **Slope of AD** (between \( A(3, 2) \) and \( D(2, 7) \)):
\[
m_{AD} = \frac{7 - 2}{2 - 3} = \frac{5}{-1} = -5
\]

3. **Slope of BC** (between \( B(13, 4) \) and \( C(12, 9) \)):
\[
m_{BC} = \frac{9 - 4}{12 - 13} = \frac{5}{-1} = -5
\]

4. **Slope of CD** (between \( C(12, 9) \) and \( D(2, 7) \)):
\[
m_{CD} = \frac{7 - 9}{2 - 12} = \frac{-2}{-10} = \frac{1}{5}
\]

### Step 2: Verify if sides are parallel
Two lines are parallel if their slopes are equal.

- Compare \( m_{AB} \) and \( m_{CD} \):
\[
m_{AB} = \frac{1}{5}, \quad m_{CD} = \frac{1}{5}
\]
Thus, \( AB \parallel CD \).

- Compare \( m_{AD} \) and \( m_{BC} \):
\[
m_{AD} = -5, \quad m_{BC} = -5
\]
Thus, \( AD \parallel BC \).

### Step 3: Verify if sides are perpendicular
Two lines are perpendicular if the product of their slopes is \( -1 \).

- Check if \( AB \) and \( AD \) are perpendicular:
\[
m_{AB} \times m_{AD} = \frac{1}{5} \times (-5) = -1
\]
Thus, \( AB \perp AD \).

This confirms that sides \( AB \) and \( AD \) are perpendicular. Similarly, since the opposite sides are parallel and adjacent sides are perpendicular, the points form a rectangle.

### Conclusion:
The points \( A(3, 2) \), \( B(13, 4) \), \( C(12, 9) \), and \( D(2, 7) \) are vertices of a rectangle because opposite sides are parallel, and adjacent sides are perpendicular.

Would you like more details or have any questions?

### Relative questions:
1. How do you calculate the distance between two points in geometry?
2. Can the same method be used to verify if points form a square?
3. What is the significance of the slopes being equal for parallel lines?
4. How do you confirm that a quadrilateral is a parallelogram using slopes?
5. What are the conditions to check if a quadrilateral is a rhombus?

### Tip:
Always verify perpendicularity by checking if the product of slopes is \( -1 \), which confirms the right angle between two sides.

Use slopes to show that A(3, 2), B(13, 4), C(12, 9), and D(2, 7) are vertices of a rectangle. We first plot the points to find the pairs of points that determine each side. Next, we find the slopes of the four sides. The slope of AB is , the slope of AD is , the slope of BC is , and the slope of CD is . Two sides are parallel if their slopes are . Thus, we see that AB and are parallel to each other and AD and are parallel to each other. Two lines are perpendicular to one another when the product of their slopes is equal to . Since (slope of AB) ✕ (slope of AD) = , the sides AB and AD are perpendicular to each other. Therefore, the points A, B, C, and D are vertices of a rectangle.

Learn how to verify that the points A(3, 2), B(13, 4), C(12, 9), and D(2, 7) form a rectangle using slope calculations. This problem covers essential geometry concepts, such as slopes of lines, parallel and perpendicular line conditions, and is suitable for students in Grades 9-11.

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