1. **Stating the problem:** We have two questions. - First, to find the equation representing the line of reflection for the given graph, which involves polygons reflected across a vertical line passing through $x=-2$. - Second, to identify the shape formed by the points $(-1,3)$, $(2,1)$, $(2,-1)$, and $(-1,-2)$. 2. **First question: Line of reflection** - The line of reflection is given as the vertical line passing through $x=-2$. - A vertical line through $x = a$ is represented by the equation $x = a$. - Therefore, the equation for the line of reflection is $$x = -2$$ 3. **Second question: Shape from the given points** - The points are $A(-1,3)$, $B(2,1)$, $C(2,-1)$, and $D(-1,-2)$. - To identify the shape, calculate the lengths of all sides using the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ - Calculate lengths: - $AB = \sqrt{(2 - (-1))^2 + (1 - 3)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}$ - $BC = \sqrt{(2 - 2)^2 + (-1 - 1)^2} = \sqrt{0 + (-2)^2} = \sqrt{4} = 2$ - $CD = \sqrt{(-1 - 2)^2 + (-2 - (-1))^2} = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}$ - $DA = \sqrt{(-1 - (-1))^2 + (3 - (-2))^2} = \sqrt{0 + 5^2} = 5$ - Check opposite sides: - $AB \neq CD$ and $BC \neq DA$, so it is not a parallelogram. - Calculate slopes of sides to check if any sides are parallel: - Slope $AB = \frac{1 - 3}{2 - (-1)} = \frac{-2}{3}$ - Slope $BC = \frac{-1 - 1}{2 - 2} = \frac{-2}{0}$ undefined (vertical) - Slope $CD = \frac{-2 - (-1)}{-1 - 2} = \frac{-1}{-3} = \frac{1}{3}$ - Slope $DA = \frac{3 - (-2)}{-1 - (-1)} = \frac{5}{0}$ undefined (vertical) - Opposite sides $BC$ and $DA$ are both vertical (slope undefined), so these sides are parallel. - Since only one pair of opposite sides is parallel, the shape is a trapezoid. **Final answers:** - Equation of the line of reflection: $$x = -2$$ - The shape formed by the points is a **trapezoid**.