Question

select the graph of the trapezoid abcd after a reflection in the line y =x. * graph of trapezoid abcd on coordinate grid with axes x and y, points a, b, c, d marked

Explanation:

Step1: Identify coordinates of vertices

First, we find the coordinates of the vertices of trapezoid \(ABCD\). From the graph:

• \(A\): Let's assume the grid has each square with side length 1. Looking at the position, \(A\) is at \((-3, 1)\) (since it's 3 units left on the x - axis and 1 unit up on the y - axis).
• \(B\): \(B\) is at \((-2, 3)\) (2 units left on x - axis, 3 units up on y - axis).
• \(C\): \(C\) is at \((-1, 3)\) (1 unit left on x - axis, 3 units up on y - axis).
• \(D\): \(D\) is at \((0, 1)\) (on the y - axis, 1 unit up on y - axis).

Step2: Apply reflection over \(y = x\)

The rule for reflecting a point \((x,y)\) over the line \(y=x\) is \((x,y)\to(y,x)\).

• For \(A(-3,1)\): After reflection, \(A'\) is \((1, - 3)\).
• For \(B(-2,3)\): After reflection, \(B'\) is \((3, - 2)\).
• For \(C(-1,3)\): After reflection, \(C'\) is \((3, - 1)\).
• For \(D(0,1)\): After reflection, \(D'\) is \((1, 0)\).

Step3: Analyze the reflected figure

Now, we can plot these reflected points \((1, - 3)\), \((3, - 2)\), \((3, - 1)\), and \((1, 0)\) on the coordinate plane. The shape formed by these points will be the reflection of trapezoid \(ABCD\) over \(y = x\). The key is that the original trapezoid has a horizontal top and bottom (parallel to the x - axis) between \(y = 1\) and \(y = 3\) (for the non - vertical sides). After reflection over \(y=x\), the sides will be parallel to the y - axis (since the original sides were parallel to the x - axis), and the figure will be a trapezoid with vertices at the reflected coordinates.

Answer:

The graph of the trapezoid \(ABCD\) after reflection over \(y = x\) will have vertices at \((1, - 3)\), \((3, - 2)\), \((3, - 1)\), and \((1, 0)\) (the actual graph should be plotted with these points, and it will be a trapezoid with the top and bottom sides parallel to the y - axis, and the legs connecting the appropriate points).