Section 8-2 Similar Polygons SPI 31A: identify corresponding parts of congruent geometric figuresSPI 32C: determine congruence or relations between triangles or quadrilaterals given a diagramSPI 22B: apply ratio and proportion to solve real-world problems

Objectives:• identify and determine missing measurements of polygons• apply proportion to the real-world

Similar (~): • two figures have the same shape but not necessarily the same size.• Figures are ~ if:

(1) corresponding s are (2) corresponding sides are proportional

Similarity Ratio: ratio of the lengths of corresponding sides

Understanding Similarity

ABCD ~ EFGH

ABEF

= AD ?

Since the two figures are similar:

What is the mE? mE = m A = 53º

Complete the statement for corresponding sides: EH

Determine Similarity

Determine whether the triangles are similar. If they are write a similarity statement and give the similarity ratio.

Conditions for similarity: 1. corresponding s are

2. corresponding sides are proportional

ACFD

= 18 24

3 4

= ABFE

= 15 20

3 4

= BCED

= 12 16

3 4

=

∆ABC ~ ∆FED with a similarity ratio of ¾ or 3:4

Real-World Connection: Drawing

You want to draw an enlargement of a design that is painted on a 3in by 5in card. You plan to draw on a 8 ½ in by 11 in piece of paper. What are the dimensions of the largest complete enlargement you can draw?

3 in5 in

= x11

Postcard Paper

WidthLength

Write and solve a proportion

5x = 33 x = 6.6

The largest complete enlargement you can draw is 6.6 in by 11 in.

Golden Rectangle and Phi Ø

The golden rectangle is considered pleasing to the human eye. It is used extensively in architecture and art.

In any golden rectangle, the length and the width are in the golden ratio which is about 1.618 : 1.

Total 13 units

8 u

nit

s

8 units 5 units 5 units

Golden ratio = 13 1.6 8

5 u

nit

s3

un

its

Golden ratio = 8 1.6 5

Make a square

Bottom and top of the oval: Bottom to naval: 12 units Top to naval: 7.5 units

Golden ratio: 12:7.5 or 1.6

Bottom of figure to top of head Bottom to naval: 9.5 units Top to naval: 5.8 units

Golden ratio: 9.5 : 5.8 or 1.64

The golden rectangle and golden ratio depicted in

Art