To determine the type of quadrilateral formed by the points A(-6, 4), B(-3, 5), C(0, 2), and D(-3, 1), we can find the lengths of the sides and the slopes of the segments formed by these points. Additionally, we can check the angles to understand the properties of the quadrilateral.
-
Calculate the lengths of the sides (using the distance formula): The distance formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
-
\(AB\): \[ AB = \sqrt{((-3) - (-6))^2 + (5 - 4)^2} = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \]
-
\(BC\): \[ BC = \sqrt{(0 - (-3))^2 + (2 - 5)^2} = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
-
\(CD\): \[ CD = \sqrt{((-3) - 0)^2 + (1 - 2)^2} = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]
-
\(DA\): \[ DA = \sqrt{((-6) - (-3))^2 + (4 - 1)^2} = \sqrt{(-3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
-
-
Summarize the side lengths:
- \(AB = \sqrt{10}\)
- \(BC = 3\sqrt{2}\)
- \(CD = \sqrt{10}\)
- \(DA = 3\sqrt{2}\)
This shows that \(AB = CD\) and \(BC = DA\), indicating that opposite sides are equal.
-
Calculate the slopes of the sides to analyze angles: The slope formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
-
Slope of \(AB\): \[ m_{AB} = \frac{5 - 4}{-3 - (-6)} = \frac{1}{3} \]
-
Slope of \(BC\): \[ m_{BC} = \frac{2 - 5}{0 - (-3)} = \frac{-3}{3} = -1 \]
-
Slope of \(CD\): \[ m_{CD} = \frac{1 - 2}{-3 - 0} = \frac{-1}{-3} = \frac{1}{3} \]
-
Slope of \(DA\): \[ m_{DA} = \frac{4 - 1}{-6 - (-3)} = \frac{3}{-3} = -1 \]
-
-
Analysis of the slopes:
- The slopes of \(AB\) and \(CD\) are equal (\(\frac{1}{3}\)), indicating that these sides are parallel.
- The slopes of \(BC\) and \(DA\) are equal (-1), indicating that these sides are also parallel.
Since both pairs of opposite sides are equal and parallel, this confirms that the quadrilateral is a parallelogram.
- Conclusion: Since the quadrilateral has opposite sides that are equal and parallel, we classify the quadrilateral as a parallelogram.
Thus, the quadrilateral formed by the points A(-6, 4), B(-3, 5), C(0, 2), and D(-3, 1) is a parallelogram.