Class Notes Unit 7

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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

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

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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

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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:

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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

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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)

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7.4. Similar Triangles

See connect p. 344

Note the properties of similar triangles

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 500

angles. Since the angles of a triangle sum to 1800_{the third}

angle must be 800

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

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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

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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???

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)

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7.7. Identifying Types of Symmetry on the Cartesian Plane

See connect p. 369 on overhead

Complete p. 373 #3-5 together