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). (from unit 1, lesson 3.)

Explanation:

2.

Step1: Observe orientation change

Notice that trapezoid A needs to be rotated. Assume a 90 - degree counter - clockwise rotation about a point (for example, the origin if on a coordinate plane). Let the vertices of trapezoid A be \((x,y)\). After a 90 - degree counter - clockwise rotation about the origin, the transformation rule is \((x,y)\to(-y,x)\).

Step2: Observe position change

After rotation, trapezoid A needs to be translated. Count the number of units it needs to move horizontally and vertically to match trapezoid B. Let's say it needs to move \(a\) units to the right and \(b\) units down. The translation rule for a point \((x,y)\) is \((x,y)\to(x + a,y - b)\).

Step1: Identify vertices of polygon P

Let the vertices of polygon P be \(P_1(x_1,y_1),P_2(x_2,y_2),\cdots\).

Step2: Use reflection formula

For a point \((x,y)\) reflected over a vertical line \(x = k\) (assuming line \(\ell\) is a vertical line \(x = k\)), the reflection formula is \((x,y)\to(2k - x,y)\). For a horizontal line \(y=k\), the reflection formula is \((x,y)\to(x,2k - y)\). Measure the distance of each vertex of polygon P from line \(\ell\) and find the corresponding reflected point on the other side of line \(\ell\) at the same distance.

Step3: Connect reflected points

Connect the reflected vertices in the same order as the original polygon P to form the reflected polygon.

Answer:

First, rotate trapezoid A 90 - degree counter - clockwise about a suitable point. Then, translate the rotated trapezoid to the right and down by appropriate units.

3.