### Unit 7: Similarity and Transformations

Students need rulers & grid paper! Accuracy when measuring is very important.

**7.1. Scale Drawings and Enlargements**

An **enlargement** occurs when an object is made bigger.

A **reduction** occurs when an object is made smaller.

See investigate p. 318 on overhead

See connect p. 319 on overhead – note terminology:

Scale diagram Proportion

Corresponding lengths Scale factor

IMPORTANT: Enlargements (where the drawing is bigger than the original) are always associated with a scale factor greater than 1.

See examples 1-3 pp. 320-322 on overheads

Note: Scale factors can also be written as percents.

Other example. Some viruses measure 0.0001 mm in diameter. An artist’s diagram of a virus shows the diameter as 5 mm.

Determine the scale factor used.

Solution:

Scale factor = scale length = 5 = 50000 original length 0.0001

Discuss p. 322 #3

Complete p. 323 #4 & 5 together on overhead Discuss angles in #9 p. 323

**7.2. Scale Drawings and Reductions**

See connect p. 326 on overhead

IMPORTANT: Reductions (where the drawing is smaller than the original) are always associated with a scale factor less than 1.

See examples 1 & 2 pp. 326-328 on overhead

A map is a great example of a scale drawing that represents a reduction from the original. In such cases the scale is usually given as a ratio. Note that the ratio is always written as:

Drawing : Actual

just as the scale factor is always given as: Drawing ÷ Actual

Example. If the scale on a map is 1:500000,

(a) How many kilometers does 7.5cm represent?

Solution:

7.5 x 500000 = 3750000cm = 37500m = 37.5 km

(b) How many cm on the map would represent and actual distance of 25 km?

Solution:

25 ÷ 500000 = 0.00005km = 0.05m = 5 cm

Note:

Scale factor = image length ÷ original length Image length = scale factor x original length Original length = image length ÷ scale factor

Complete the following table:

**Scale Factor** **Original Length** **Image Length**

4 6cm

0.6 30cm

25% 160m

18m 6m

Complete p. 329 #4-6 together orally

Set pp. 329-331 #8-15, 20 (need grid paper for #12)

Journal:

**7.3. Similar Polygons**

Investigation pp. 334

The original figure is quadrilateral ABCD. Assume we are using 0.5cm grid paper

Quadrilateral ABCD has been enlarged by a scale factor of 3 to produce A’B’C’D’. Look at the table that follows:

Lengths of side (cm)

AB A’B’ BC B’C’ CD C’D’ DA D’A’

2 6 3 9 2.8 8.4 1.7 5.1

Measures of angle
(o_{)}

A A’ B B’ C C’ D D’

118 118 90 90 72 72 82 82

Quadrilateral ABCD has been reduced by a scale factor of 0.5 to produce A’’B’’C’’D’’. Look at the table that follows:

Lengths of side (cm)

AB A’’B’’ BC B’’C’’ CD C’’D’’ DA D’’A’’

2 1 3 1.5 2.8 1.4 1.7 0.85

Measures of angle
(o_{)}

A A’

’

B B’’ C C’’ D D’’

118 118 90 90 72 72 82 82

What do you notice about the measure of the matching/corresponding angles in both tables above?

**Matching/corresponding angles are equal**

Complete the table below:

AB

A’B’ B’C’BC C’D’CD D’ADA A’’B’’AB B’’C’’BC C’’D’’CD D’’A’’DA

What do you notice about the ratios of matching sides?

**The ratio of matching/corresponding sides are equal**

See connect pp. 335-336 noting terminology: Similar

Corresponding angles Corresponding sides

**Note the properties of similar polygons on p. 336**

See examples 1-4 pp. 337-340

Complete p. 341 #4-8 together (need grid paper for #7 & isometric grid paper for #8)

**7.4. Similar Triangles**

See connect p. 344

Note the properties of similar triangles

See examples 1-4 pp. 345-349

Other Example: One triangle has two 500_{ angles. Another has a 50}0_{ angle }

and an 800_{ angle. Could the triangles be similar?}

**The triangles may be similar since all corresponding angles are ****equal. We know this because the first triangle has two 50****0**

**angles. Since the angles of a triangle sum to 180****0**_{ the third }

**angle must be 80****0**

Complete p. 349 #4 & 5 together

Set pp. 349-350 #6, 7, 9, 10, 11, 12

Set Mid-Unit Review p. 352 & Quiz or Assignment

**7.5. Reflections and Line Symmetry**

A 2-D shape has line or reflective symmetry if one half of the shape is a
reflection of the other half. The reflection occurs across a line, called the
**line of symmetry, or line of reflection. It can be horizontal, vertical or **
oblique, and may or may not be part of the diagram itself. There may be
more than one line of symmetry.

See the diagrams in Investigation p. 353
**(Provide photocopies for students)**

See other examples of letters, pictures, logos, etc. including p. 357 #3

Reflectional Symmetry and Regular Polygons

Complete the following table & draw the lines of symmetry on the regular polygons provided

Reflectional Symmetry in Regular Polygons

Number of sides & vertices 3 4 5 6 8 … n Number of Reflectional

Symmetries

3 4 5 6 8 … n

Conclusion: *The number of reflectional symmetries in an n-sided regular *
*polygon is n, the number of sides.*

See connect p. 354

See examples 1-3 pp. 354-357

**7.6. Rotations and Rotational Symmetry**

Recall that a **rotation** is a transformation, also referred to as a turn.

See the shapes in investigation p. 361

See connect p. 362 and note terminology: Rotational symmetry

Order of rotation

Angle of rotational symmetry

Note that the order of rotational symmetry of a regular polygon is the same as the number of vertices.

See letters of the alphabet

Make Snowflakes here???

See examples 1-3 pp. 362-365

Note: If a shape has order infinity it must be a circle

Complete p. 365 #4-6 together

Set pp. 365-367 #7-12, 14, 15

**(need grid paper for #9, 12, 14, 15; need isometric dot paper for #10)**

**7.7. Identifying Types of Symmetry on the Cartesian Plane**

See connect p. 369 on overhead

See examples 1-3 pp. 369-372

Complete p. 373 #3-5 together