(B) If \( \mathrm{A}(4,3), B(7,0), \mathrm{C}(1,-2) \) and \( \mathrm{D}(1,2) \) Prove that : (1) \( \overline{\mathrm{AD}} / / \overline{\mathrm{BC}} \) (2) The figure ABCD is trapezoid

To prove that lines \( \overline{\mathrm{AD}} \) and \( \overline{\mathrm{BC}} \) are parallel, we first calculate the slopes of these lines. The slope of a line given two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is determined by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). 1. For \( \overline{\mathrm{AD}} \) with points \( A(4,3) \) and \( D(1,2) \): \[ m_{AD} = \frac{2 - 3}{1 - 4} = \frac{-1}{-3} = \frac{1}{3} \] 2. For \( \overline{\mathrm{BC}} \) with points \( B(7,0) \) and \( C(1,-2) \): \[ m_{BC} = \frac{-2 - 0}{1 - 7} = \frac{-2}{-6} = \frac{1}{3} \] Since both lines have the same slope (\( \frac{1}{3} \)), we can conclude that \( \overline{\mathrm{AD}} \parallel \overline{\mathrm{BC}} \). Next, to establish that figure ABCD is a trapezoid, we recall that a trapezoid is defined as a quadrilateral with at least one pair of parallel sides. Since we have shown \( \overline{\mathrm{AD}} \parallel \overline{\mathrm{BC}} \), ABCD meets the criteria for being a trapezoid. Enjoy exploring the geometry around you! Knowing how to identify figures like trapezoids through their properties can not only enhance your math skills but can also help you understand architecture, art, and even nature, where these shapes appear! If you're keen on diving deeper into geometry, there are fantastic resources such as "Geometry for Dummies" or even online platforms like Khan Academy that offer interactive lessons on the subject. It's a great way to brush up on those angles and figures!