: Similar Polygons
- The Student Will investigate similar figures
- The Student Will apply properties of proportions to similar figures
. Basic geometry of polygons; knowledge of quadrilaterals.
: Draw a number of pentagons--some regular, some irregular. Ask the
students to group those that are alike--what reasoning did they use in
grouping them together?
Elicit the primary characteristic of similarity from the motivational problem--namely, that the regular pentagons all appear to be dilations
of each other. That is, they have been magnified or reduced in some constant fashion. The other pentagons may have five sides, but they are not images of each other.
The idea of dilation
is the key to similarity, but the nuts and bolts of similarity are--congruent corresponding angles and corresponding proportional sides. Questions such as "Are all squares similar? Are all rectangles similar?" are excellent tools to debunk naive notions such as angle congruence is sufficient. Appropriately constructed trapezoids can be helpful in demonstrating the corresponding
aspect of proportional sides.
In practice, make sure students can identify scale factor
and understand that the order in which the figures are considered determines whether you consider k
as the scale factor.
In addition to plenty of straight-forward practice identify and applying scale factors, students comfortable with coordinate geometry can investigate dilations in the plane and even prove--using the distance formula--that dilation by a factor of k
increases all lengths by a factor of k
: Compare and contrast similarity and congruence. What makes two figures similar?
Triangles will be the focus of much of the study of similarity--so situations involving parallel lines and included (or vertical) angles will be frequently encountered.