Geometry: An Interactive Journey to Mastery

Lesson 30: The Mathematics of Symmetry

Solution

Assume that the dilation is centered about the origin O and has

scale factor k. Suppose that the three collinear points A, B, and C

are mapped to positions ABcc c,,and,C respectively. Our goal is to

prove that ABcc c,,andC are collinear. Mark the angles a and b as

shown. We have that a + b ƒͼ௘6HHFigure 30.3௘ͽ

We have that OAc ˜k OA, OBc ˜k OB, and OCc ˜k OC

It follows that ++AOBcc~ AOBE\WKH6$6SULQFLSOHͼ௘6LGHVFRPHLQWKHVDPHUDWLRk, and the triangles share a

common angle at O௘ͽ

Thus, ‘ ABO acc , because they are matching angles in similar triangles.

Similarly, ++BOCcc~ BOC and ‘ OB Cccb.

Thus, ‘ ‘‘  ABCccc ccABO OBCcca b 180°, which shows that ABcc c,,andC are indeed collinear.

&RPPHQW<RXFDQVKRZLQJHQHUDOWKDWDGLODWLRQPDSVDQ\VHWRIFROOLQHDUSRLQWVWRFROOLQHDUSRVLWLRQV,Q

particular, dilations map straight line segments to straight line segments.

Study Tip

x This lesson is purely optional. There are no recommended study tips for this lesson other than to enjoy

the lesson and let the thinking about it strengthen your understanding of geometry as a whole.

Pitfall

x Don’t forget to have fun in your thinking of mathematics. This is a fun topic.

1. ([SODLQZK\DUHÀHFWLRQLVDQLVRPHWU\


2. Which regular polygons have 180° rotational symmetry?

ab

O

A

Aƍ

B

Bƍ

C

Cƍ

Figure 30.3

Problems