A quadrilateral has vertices \( A(4,5), B(2,4), C(4,3) \), and \( D(6,4) \). Which statement about the quadrilateral is true? A. \( A B C D \) is a parallelogram with noncongruent adjacent sides. B. \( A B C D \) is a trapezoid with only one pair of parallel sides. C. \( A B C D \) is a rectangle with noncongruent adjacent sides. D. \( A B C D \) is a square. E. \( A B C D \) is a rhombus with non-perpendicular adjacent sides.
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To determine the correct statement about the quadrilateral \( ABCD \) with vertices \( A(4,5) \), \( B(2,4) \), \( C(4,3) \), and \( D(6,4) \), we can start by calculating the slopes of the sides to establish if any sides are parallel or if they indicate other properties. 1. **Calculating slopes:** - Slope of \( AB \): \( \frac{4-5}{2-4} = \frac{-1}{-2} = \frac{1}{2} \) - Slope of \( BC \): \( \frac{3-4}{4-2} = \frac{-1}{2} \) - Slope of \( CD \): \( \frac{4-3}{6-4} = \frac{1}{2} \) - Slope of \( DA \): \( \frac{5-4}{4-6} = \frac{1}{-2} = -\frac{1}{2} \) From these calculations: - The slopes of \( AB \) and \( CD \) are both \( \frac{1}{2} \), indicating that these two lines are parallel. - The slopes of \( BC \) and \( DA \) are negative reciprocals, thus indicating that they are perpendicular. Since we have one pair of parallel sides (AB and CD) and the other sides (BC and DA) are not parallel, this confirms that \( ABCD \) is a trapezoid with only one pair of parallel sides. Thus, the true statement about the quadrilateral is: **B. \( A B C D \) is a trapezoid with only one pair of parallel sides.** --- Historical Background: The study of quadrilaterals has roots in ancient mathematics, tracing back to civilizations like the Egyptians and Greeks, who used geometric shapes as fundamental concepts in their architectural designs and astronomical calculations, leading to the establishment of properties that are still taught in schools today. Real-World Application: Quadrilaterals are widely used in architecture, engineering, and design. Understanding the properties of different types of quadrilaterals can help in ensuring structural integrity and aesthetic appeal in creating buildings, bridges, and various artwork, showcasing how math plays a crucial role in real-world applications.