To prove that the quadrilateral ABCD, with vertices A(2,1), B(1,3), C(-5,0), and D(-4,-2), is a rectangle, we need to demonstrate that it has four right angles and opposite sides that are equal in length. First, calculate the slopes of the sides to verify that consecutive sides are perpendicular, implying that all angles are right angles. The slope of line segment AB is given by: m_AB = (y2 - y1) / (x2 - x1) = (3 - 1) / (1 - 2) = 2 / -1 = -2. Similarly, calculate the slopes of the other sides: m_BC = (y_C - y_B) / (x_C - x_B) = (0 - 3) / (-5 - 1) = -3 / -6 = 1/2, m_CD = (y_D - y_C) / (x_D - x_C) = (-2 - 0) / (-4 - -5) = -2 / 1 = -2, m_DA = (y_A - y_D) / (x_A - x_D) = (1 - (-2)) / (2 - (-4)) = 3 / 6 = 1/2. Notice that slopes m_AB and m_CD are both -2, indicating AB and CD are parallel. Similarly, slopes m_BC and m_DA are both 1/2, indicating BC and DA are parallel. Additionally, the slopes of AB and BC, and CD and DA are negative reciprocals, implying that AB is perpendicular to BC and CD is perpendicular to DA, thus confirming that all angles are right angles. Next, verify that opposite sides are of equal length using the distance formula: d = √((x2 - x1)² + (y2 - y1)²). The length of side AB is: d_AB = √((1 - 2)² + (3 - 1)²) = √((-1)² + (2)²) = √(1 + 4) = √5. Calculate the lengths of the other sides: d_BC = √((-5 - 1)² + (0 - 3)²) = √((-6)² + (-3)²) = √(36 + 9) = √45, d_CD = √((-4 - -5)² + (-2 - 0)²) = √((1)² + (-2)²) = √(1 + 4) = √5, d_DA = √((2 - -4)² + (1 - -2)²) = √((6)² + (3)²) = √(36 + 9) = √45. The lengths of AB and CD are both √5, and the lengths of BC and DA are both √45, confirming that opposite sides are equal. Since both pairs of opposite sides are parallel and equal in length, and all four angles are right angles, quadrilateral ABCD is indeed a rectangle. To graph the rectangle, plot the points on a coordinate plane and draw lines connecting them in order, from A to B, B to C, C to D, and D back to A. Use graphing tools or software for precise visualization and to provide graphical evidence to support the analytical proof.

To prove that the quadrilateral ABCD is a rectangle, we need to show it has four right angles and opposite sides of equal length. We can use the slopes of the sides to check for right angles and the distance formula to compare side lengths.

Step 1: Calculate the slopes of the sides to determine if they are perpendicular to each other (indicating right angles). - Slope of AB: \( m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{3 - 1}{1 - 2} = \frac{2}{-1} = -2 \) - Slope of BC: \( m_{BC} = \frac{y_C - y_B}{x_C - x_B} = \frac{0 - 3}{-5 - 1} = \frac{-3}{-6} = \frac{1}{2} \) - Slope of CD: \( m_{CD} = \frac{y_D - y_C}{x_D - x_C} = \frac{-2 - 0}{-4 - (-5)} = \frac{-2}{1} = -2 \) - Slope of DA: \( m_{DA} = \frac{y_A - y_D}{x_A - x_D} = \frac{1 - (-2)}{2 - (-4)} = \frac{3}{6} = \frac{1}{2} \)

Since \( m_{AB} = m_{CD} \) (both -2) and \( m_{BC} = m_{DA} \) (both \( \frac{1}{2} \)), AB is parallel to CD and BC is parallel to DA. Also, since \( m_{AB} \) and \( m_{BC} \)'s product is -1, they are perpendicular, as are \( m_{CD} \) and \( m_{DA} \).

Step 2: Use the distance formula to calculate the lengths of opposite sides to show they are equal. - \( d_{AB} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(1 - 2)^2 + (3 - 1)^2} = \sqrt{(-1)^2 + 2^2} = \sqrt{5} \) - \( d_{BC} = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(-5 - 1)^2 + (0 - 3)^2} = \sqrt{36 + 9} = \sqrt{45} \) - \( d_{CD} = \sqrt{(x_D - x_C)^2 + (y_D - y_C)^2} = \sqrt{(-4 - (-5))^2 + (-2 - 0)^2} = \sqrt{1 + 4} = \sqrt{5} \) - \( d_{DA} = \sqrt{(x_A - x_D)^2 + (y_A - y_D)^2} = \sqrt{(2 - (-4))^2 + (1 - (-2))^2} = \sqrt{36 + 9} = \sqrt{45} \)

Since \( d_{AB} = d_{CD} \) and \( d_{BC} = d_{DA} \), opposite sides are of equal length.

As a result, ABCD has four right angles and opposite sides that are equal in length, proving that it is indeed a rectangle.

Extra: In geometry, a rectangle is defined as a quadrilateral with four right angles (each measuring 90 degrees). In addition to having right angles, a rectangle also has the property that opposite sides are parallel and of equal length, which is a consequence of the definition.

When working with coordinates and slopes in a plane: - Parallel lines have the same slope. - Perpendicular lines have slopes that are negative reciprocals of each other (meaning their product is -1).

The distance formula, which is derived from the Pythagorean theorem, allows us to find the length of a side given two endpoints. This formula is given by: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

Understanding how to use the distance formula and slope calculations is key to solving many problems in coordinate geometry, such as proving the nature of a polygon.