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
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, the slope of AD is
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, the slope of BC is
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, and the slope of CD is
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. Two sides are parallel if their slopes are
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. Thus, we see that AB and
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are parallel to each other and AD and
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are parallel to each other. Two lines are perpendicular to one another when the product of their slopes is equal to
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. Since (slope of AB) ✕ (slope of AD) =
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, 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 the points \( A(3, 2), B(13, 4), C(12, 9), D(2, 7) \) are vertices of a rectangle using slopes, we follow these steps:

### Step 1: Calculate the slopes of the sides.
The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

1. **Slope of AB** (between points \( A(3, 2) \) and \( B(13, 4) \)):

\[
\text{slope of AB} = \frac{4 - 2}{13 - 3} = \frac{2}{10} = \frac{1}{5}
\]

2. **Slope of AD** (between points \( A(3, 2) \) and \( D(2, 7) \)):

\[
\text{slope of AD} = \frac{7 - 2}{2 - 3} = \frac{5}{-1} = -5
\]

3. **Slope of BC** (between points \( B(13, 4) \) and \( C(12, 9) \)):

\[
\text{slope of BC} = \frac{9 - 4}{12 - 13} = \frac{5}{-1} = -5
\]

4. **Slope of CD** (between points \( C(12, 9) \) and \( D(2, 7) \)):

\[
\text{slope of CD} = \frac{7 - 9}{2 - 12} = \frac{-2}{-10} = \frac{1}{5}
\]

### Step 2: Check for parallel sides.
For a rectangle, opposite sides must be parallel. Two sides are parallel if their slopes are equal.

- Slope of **AB** \( = \frac{1}{5} \) and slope of **CD** \( = \frac{1}{5} \). Therefore, **AB** is parallel to **CD**.
- Slope of **AD** \( = -5 \) and slope of **BC** \( = -5 \). Therefore, **AD** is parallel to **BC**.

### Step 3: Check for perpendicular sides.
For a rectangle, adjacent sides must be perpendicular. Two lines are perpendicular if the product of their slopes is \( -1 \).

- Slope of **AB** \( = \frac{1}{5} \) and slope of **AD** \( = -5 \).
  
\[
\left(\frac{1}{5}\right) \times (-5) = -1
\]

This confirms that **AB** is perpendicular to **AD**.

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

Would you like any further details or have any questions?

### Related Questions:
1. How can you use the distance formula to further verify this is a rectangle?
2. How do you verify if four points form a square instead of a rectangle?
3. What happens if the slopes are equal but the points don't form a closed shape?
4. Can you generalize this method for any quadrilateral to check its type?
5. How do you calculate the area of this rectangle using the given points?

### Tip:
When using slopes to determine perpendicularity, always remember that the product of the slopes of two perpendicular lines is \( -1 \).

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.

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 solution uses the slope formula to prove that opposite sides are parallel and adjacent sides are perpendicular. Suitable for Grades 9-12 students studying coordinate geometry.

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