Question

1. describe a sequence of transformations that takes trapezoid a to trapezoid b. (from unit 1, lesson 4.) 3. reflect polygon p using line (ell).

Explanation:

Step 1: Analyze orientation change

Trapezoid A needs to be rotated to match the orientation of trapezoid B. Assume a 90 - degree counter - clockwise rotation about a point. Let's say we rotate it 90 degrees counter - clockwise about the origin (if we consider a coordinate system). The general rule for a 90 - degree counter - clockwise rotation of a point $(x,y)$ about the origin is $(x,y)\to(-y,x)$.

Step 2: Analyze position change

After rotation, trapezoid A needs to be translated to the position of trapezoid B. Let the translation vector be $(a,b)$. We find the difference in the $x$ and $y$ coordinates of a corresponding vertex of the rotated trapezoid A and trapezoid B to determine $a$ and $b$.

For question 3:

Step 1: Identify key points

For polygon P, identify all the vertices. Let the vertices be $V_1(x_1,y_1),V_2(x_2,y_2),\cdots$.

Step 2: Reflect vertices

For each vertex $V_i(x_i,y_i)$ of polygon P, to reflect it over the line $\ell$ (assuming $\ell$ is a vertical line with equation $x = k$), the new $x$ - coordinate is $2k - x_i$ and the $y$ - coordinate remains the same, i.e., the new vertex is $(2k - x_i,y_i)$. Connect the reflected vertices to get the reflected polygon.

Answer:

1. A 90 - degree counter - clockwise rotation about a suitable point followed by a translation.
2. Reflect each vertex of polygon P over line $\ell$ using the rule for reflection over a vertical line (or horizontal line depending on $\ell$) and then connect the reflected vertices.