For Exercises \( 7-10 \), graph each quadrilateral using the given vertices. Then use the distance formula and the slope formula to determine the most specific name for each quadrilateral: trapezoid, kite, rectangle, rhombus, square, parallelogram, or just quadrilateral. \( \begin{array}{ll}\text { 7. } A(6,8), B(9,7), C(7,1), D(4,2) & \text { 8. } E(1,-2), F(5,-5), G(2,-8) \text {, }\end{array} \)

To determine the most specific name for each quadrilateral given the vertices, we will follow these steps: 1. **Graph the quadrilaterals using the given vertices.** 2. **Calculate the lengths of the sides using the distance formula.** 3. **Calculate the slopes of the sides using the slope formula.** 4. **Analyze the results to classify the quadrilaterals.** ### Quadrilateral 7: Vertices \( A(6,8), B(9,7), C(7,1), D(4,2) \) #### Step 1: Graph the Quadrilateral Plot the points \( A(6,8) \), \( B(9,7) \), \( C(7,1) \), and \( D(4,2) \) on a coordinate plane. #### Step 2: Calculate the Lengths of the Sides Using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] - Length \( AB \): \[ d_{AB} = \sqrt{(9 - 6)^2 + (7 - 8)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \] - Length \( BC \): \[ d_{BC} = \sqrt{(7 - 9)^2 + (1 - 7)^2} = \sqrt{(-2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \] - Length \( CD \): \[ d_{CD} = \sqrt{(4 - 7)^2 + (2 - 1)^2} = \sqrt{(-3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \] - Length \( DA \): \[ d_{DA} = \sqrt{(6 - 4)^2 + (8 - 2)^2} = \sqrt{(2)^2 + (6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \] #### Step 3: Calculate the Slopes of the Sides Using the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] - Slope \( AB \): \[ m_{AB} = \frac{7 - 8}{9 - 6} = \frac{-1}{3} \] - Slope \( BC \): \[ m_{BC} = \frac{1 - 7}{7 - 9} = \frac{-6}{-2} = 3 \] - Slope \( CD \): \[ m_{CD} = \frac{2 - 1}{4 - 7} = \frac{1}{-3} = -\frac{1}{3} \] - Slope \( DA \): \[ m_{DA} = \frac{8 - 2}{6 - 4} = \frac{6}{2} = 3 \] #### Step 4: Analyze the Results - The lengths of opposite sides are equal: \( AB = CD \) and \( BC = DA \). - The slopes of opposite sides are equal: \( m_{AB} = m_{CD} \) and \( m_{BC} = m_{DA} \). Since both pairs of opposite sides are equal in length and parallel, this quadrilateral is a **parallelogram**. ### Quadrilateral 8: Vertices \( E(1,-2), F(5,-5), G(2,-8) \) #### Step 1: Graph the Quadrilateral Plot the points \( E(1,-2) \), \( F(5,-5) \), and \( G(2,-8) \) on a coordinate plane. Note that this is a triangle, not a quadrilateral. #### Step 2: Calculate the Lengths of the Sides - Length \( EF \): \[ d_{EF} = \sqrt{(5 - 1)^2 + (-5 + 2)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] - Length \( FG \): \[ d_{FG} = \sqrt{(2 - 5)^2 + (-8 + 5)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \] - Length \( GE \): \[ d_{GE} = \sqrt{(1 - 2)^2 + (-2 + 8)^2} = \sqrt{(-1)^2 + (6)^2} = \sqrt{1 + 36} = \sqrt{37} \] #### Step 3: Calculate the Slopes of the Sides - Slope \( EF \): \[ m_{EF} = \frac{-5 + 2}{5 - 1} = \frac{-3}{4} \] - Slope \( FG \): \[ m_{FG} = \frac{-8 + 5}{2 - 5} = \frac{-3}{-3} = 1 \] - Slope \( GE \): \[ m_{GE} = \frac{-2 + 8}{1 - 2} = \frac{6}{-1} = -6 \] #### Step 4: Analyze the Results - The lengths of the sides are not equal. - The slopes are not equal, indicating no parallel sides. Since this shape is a triangle and does not fit any of the quadrilateral classifications, it is simply a **triangle**. ### Summary 1. Quadrilateral 7 is a **parallelogram**. 2. Quadrilateral 8 is a **triangle** (not a quadrilateral).