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 $-1$.