Friday, September 19, 2014

Chapter 4 Quiz (Day 31)

Here is my version of the Chapter 4 Quiz. In this post I explain the answers.

1. Students should draw a point P' on the other side of line m, the same distance as P, but in the opposite direction. In other words, P' should be drawn so that m is the perpendicular bisector of PP'.

2. Students should draw the perpendicular bisector of PP'. But notice that we haven't covered constructions yet, so students should just estimate its position.

3. The measure of the angle is 48 degrees -- exactly double that of the given angle.

4. The symmetry line is the line containing the angle bisector of the given angle. This follows directly from the Angle Symmetry Theorem. The hard part about writing this quiz is that I couldn't include every single property or theorem on the Quiz Review. Students should be familiar with all of the properties of reflections (in other words, the Reflection Postulate) as well as of the results derived from these properties (the theorems).

5. The orientation is counterclockwise, of course, since reflections switch orientation.

6. Reflections preserve distance.

7. There are two pairs of angles -- angles B and C, and angles BAD and CAD.

8. The conclusion follows from the definition of symmetry line and the fact that reflections preserve angle measure. Technically, we need the definition of symmetry line -- we can't just use the Reflection Postulate directly because nowhere in the picture is any mention of a reflection. So we need the definition of symmetry line to get from "line AD is a symmetry line" to "the reflection image of triangle ABC is triangle ACB." But I'd accept it as a correct answer if a student only mentioned that reflections preserve angle measure, since a full two-column proof isn't required.

9. A square has four symmetry lines -- one horizontal, one vertical, two diagonal. This is actually not proved until later on -- but once again, a full proof isn't required.

10. The reflection image is (-a, b). There were actually examples of this in the Exercises, but only where the coordinates were numbers, not variables. If the students are confused, a teacher can change this question to something like the reflection image of (1, 2), and even encourage the students to draw in a quick graph to demonstrate it.

Hopefully I didn't make this quiz too difficult for the students.

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