Question

polygon abcd reflects about line mn to make polygon efgh. what is the length of fg? a. 2 units b. 2.2 units c. 3 units d. 3.2 units

Explanation:

Step1: Recall reflection property

Reflection is a rigid transformation, so corresponding sides of the original polygon and its image are congruent.

Step2: Identify corresponding sides

In polygon \(ABCD\) and its reflection \(EFGH\), side \(AB\) corresponds to \(EF\), \(BC\) corresponds to \(FG\)? Wait, no, let's check the vertices. Wait, when reflecting over line \(MN\), the correspondence: \(A\) reflects to \(E\), \(B\) reflects to \(F\), \(C\) reflects to \(G\), \(D\) reflects to \(H\)? Wait, no, looking at the diagram, \(AB\) is vertical with length 3, \(BC\) is horizontal with length 2, \(CD\) is length 2.2, \(DA\) is 3.2. Then \(FG\) should correspond to \(BC\)? Wait, no, maybe \(AB\) corresponds to \(EF\), \(BC\) corresponds to \(FG\)? Wait, no, let's see the direction. Wait, actually, in reflection, the length of corresponding sides are equal. So \(FG\) should be equal to \(BC\)? Wait, no, \(AB\) is length 3, \(BC\) is length 2, \(CD\) is 2.2, \(DA\) is 3.2. Wait, maybe \(FG\) corresponds to \(AB\)? Wait, no, let's check the coordinates. Wait, the original polygon \(ABCD\): \(B\) to \(C\) is length 2 (horizontal), \(A\) to \(B\) is length 3 (vertical). Then the reflected polygon \(EFGH\): \(F\) to \(G\) should correspond to \(A\) to \(B\)? Wait, no, maybe I got the correspondence wrong. Wait, the key is that reflection preserves length, so the side \(FG\) in \(EFGH\) should be equal to the corresponding side in \(ABCD\). Looking at the options, and the sides of \(ABCD\): \(AB = 3\), \(BC = 2\), \(CD = 2.2\), \(DA = 3.2\). Now, \(FG\) in \(EFGH\) – let's see, the side \(AB\) is vertical with length 3, so if \(FG\) is vertical, then \(FG\) should be equal to \(AB\), which is 3? Wait, no, the options have 3 as option C. Wait, but let's re-examine. Wait, maybe \(FG\) corresponds to \(AB\). Wait, the original \(AB\) is length 3, so \(FG\) should be 3. Wait, but let's check the correspondence again. When reflecting over line \(MN\), the image of \(B\) is \(F\), image of \(A\) is \(E\), image of \(C\) is \(G\), image of \(D\) is \(H\). So \(AB\) (from \(A\) to \(B\)) reflects to \(EF\) (from \(E\) to \(F\)), \(BC\) (from \(B\) to \(C\)) reflects to \(FG\) (from \(F\) to \(G\))? Wait, no, \(B\) to \(C\) is horizontal, \(F\) to \(G\) is vertical? Wait, maybe I mixed up the correspondence. Wait, the length of \(AB\) is 3 (vertical), so if \(FG\) is vertical, then \(FG = AB = 3\). So the length of \(FG\) is 3 units.

Answer:

C. 3 units