September 2, 2008

A collection of lessons for a unit on Similarity.

- Education > General
- Mathematics > General
- Mathematics > Geometry
- Mathematics > General
- Mathematics > Geometry

- Grade 9
- Grade 10
- Grade 11
- Grade 12

**Topic**: Similar Polygons

**Learning Objectives:**

*The Student Will*investigate similar figures*The Student Will*apply properties of proportions to similar figures

**Motivational Problem**: 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?

**Overview**

Elicit the primary characteristic of similarity from the motivational problem--namely, that the regular pentagons all appear to b*e 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* or *1/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*

**Key Questions**: Compare and contrast similarity and congruence. What makes two figures similar?

**Looking Ahead:** 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.

**Topic**: Similar Triangles

**Learning Objectives:**

*The Student Will*apply properties of similarity to triangles*The Student Will*utilize similarity in geometric proofs

**Motivational Problem**: Draw some examples of simiiar triangles. What do all these triangles have in common? What would you need to know, at minimum, to show that two triangles were similar?

**Overview**

Depending on how you choose to develop the topic of triangle similarity, you may accept as a postulate either (or neither) Angle-Angle-Angle or Angle-Angle. If the former, the latter can be established immediately as a theorem using the simple and elegant argument -- "If, in two triangles, two pairs of corresponding angles are congruent, then the third pair must be congruent."

Common situations were similar triangles occur are when transversals of parallel lines cross (creating vertical angles) and when, inside a triangle, internal segments are drawn parallel to a given side (creating corresponding angles). The SAS and SSS similarity theorems are nice analogues to the congruence theorems, with "congruent sides" being replaced by "sides in proportion".

**Key Questions**: When are two triangles similar? Give examples of situations in which similar triangles occur. Compare and contrast SSS similarity and SSS congruence.

**Looking Ahead:** A classic proof of the Pythagorean Theorem uses the similar triangles created when the altitude to the hypotenuse is drawn. Similarity is also key to the "Power of a Point" theorems in circle geometry.

**Topic**: Proportionality Theorems

**Learning Objectives:**

*The Student Will*apply similarity to develop some proportionality theorems.*The Student Will*apply proportionality in the context of parallel lines and angle bisectors.

**Motivational Problem**: Given a triangle with an internal segment parallel to a side, ask students to give an justify three true proportions for the figure.

**Overview**

Two important and immediate consequences of similarity in triangles are:

- Parallel lines cutting off proportional lengths
- The Angle Bisector Theorem

So given three (or more) parallel lines cutting across two transversals, you can easily establish that the resulting segments are in proportion--the transversals will create a trapezoid, and by connecting opposite vertices with a diagonal, you have created two triangles, and therefore two opportunities to apply the above theorem. This is a nice and natural extension of the theorem that states "If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on all transversals.

The Angle Bisector Theorem can be established by adding an auxiliary line parallel to the bisector through a different vertex of the triangle. By extending the appropriate side, the same situation as above is created, and a direct application of similarity yields the result. Interested students can look up "Mass Point Geometry" for a curious approach to Triangle angle bisectors.

**Key Questions**: In what common situations do we find similar triangles? What are some properties of the angle bisector?

**Looking Ahead:** The situation repeatedly exploited above is also a useful platform for exploring the relationship of area in similar figures, as the ratio of base-to-base is the scale-factor for the triangles, and squaring that give the proportionality constant for areas.