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.