Similarity and Transformations Math 9 – Unit 7 Similarity and Transformations

7.1 Scale Diagrams and Enlargements Concept Review At your table group read aloud and complete pages 316 and 317 from text 10-15 minutes max Key Review – isosceles characteristics, angle sum theory, Pythagoras and symmetry concepts

7.1 Scale Diagrams and Enlargements A SCALE DIAGRAM is an enlargement or a reduction of the original To find the amount an object is enlarged or reduced by we use the equation:

7.1 Scale Diagrams and Enlargements Take a look at the investigations on page 318 Matching lengths on the original diagram and on the scale diagram are called: CORRESPONDING LENGTHS So, what is the scale factor?

7.1 Scale Diagrams and Enlargements A scale factor can be expressed as a FRACTION, DECIMAL or a PERCENT. 5/2 or 2.5 or 250%

7.1 Scale Diagrams and Enlargements Pairs of corresponding sides have the SAME scale factor, so we say that corresponding sides are proportional. 5cm/2cm = 2.5 2.5cm/1cm = 2.5

7.1 Scale Diagrams and Enlargements If the scale factor below is 2, which rectangle is the original? How do you know?

7.1 Scale Diagrams and Enlargements If the scale factor is >1, 1.0 or 100% the diagram is a(n) _________________. If the scale factor is <1, 1.0 or 100% the diagram is a(n) ________________.

7.1 Scale Diagrams and Enlargements This drawing of a mosquito was printed in a newspaper and measured 4.5 cm. The actual length of the mosquito is 12mm. Determine the scale factor of the diagram. To calculate the scale factor, the units of length must be the same. What do you need to know first? 4.5 cm

7.1 Scale Diagrams and Enlargements Write this down somewhere you can reference it easily. This is information you are expected to know for grade 10 (and the grade 9 PAT).

7.1 Scale Diagrams and Enlargements What is 300 m in cm? Write 1 cm represents 5 m as a ratio. If a map scale tells you that 1 cm represents 15 km. What is 15 km in cm? Answer as a ratio. 30 000 cm 5m = 500cm so 1:500 15 km = 1,500,000cm so 1:1,500,000

7.1 Scale Diagrams and Enlargements This drawing of a mosquito was printed in a newspaper and measured 4.5 cm. The actual length of the mosquito is 12mm. Determine the scale factor of the diagram. Length of scale diagram/ length of mosquito 45 mm/ 12mm Scale factor is 3.75 4.5 cm

7.1 Scale Diagrams and Enlargements Try this: What is the scale factor? The units must be the same on the original and the scale diagram (or you have to convert them) Scale factors do not have units.

7.1 Scale Diagrams and Enlargements 5/2 = 2.5 Multiply each dimension by the scale factor. Diameter = 7.5 cm Height = 17.5 cm

7.1 Scale Diagrams and Enlargements Math Practice for HOMEWORK Page 323 – 324 Questions 4, 5, 7, 8, 11, and 14 Have these done BEFORE Monday’s math class.

7.1 – Math Practice Check Scale diagrams of different squares are to be drawn. The side length of each original and the scale factor are given. Determine the side length of each scale diagram. A – 36cm B – 205 mm C – 6.51 cm D – 171 mm E - 10

7.2 Scale Diagrams and Reductions A scale diagram can be smaller than the original diagram. This type of scale diagram is called a reduction. A reduction has a scale factor between 0 and 1.

7.2 Scale Diagrams and Reductions What is the scale factor? Can you give it as a decimal, fraction and percentage? 4/10 as a fraction 2/5 as a decimal 0/4 as a percentage

7.2 Scale Diagrams and Reductions A top view of a patio table is 105 cm by 165cm. A reduction is to be drawn with scale factor of 1/5. Find the dimensions of the reduction. Remember Tuesday’s hint…… Write scale factor as a decimal – 1/5 = 0.2 Multiple the original width and original length by 0.2 105 x 2 = 21 cm and 165 x 0.2 = 33cm

7.2 Scale Diagrams and Reductions Which diagram has sides that are proportional to the original?

7.2 Scale Diagrams and Reductions Proportion means that 2 ratios are equal. Two diagrams are proportional if ALL sides are multiplied or divided by the SAME number.

7.2 Scale Diagrams and Reductions Which diagram has sides that are proportional to the original? Original is 5 by 10, write as a fraction 5/10 and reduce to ½ 1 by 5 = 1/5 which does NOT equal ½ 2 by 6 = 1/3 which does NOT equal ½ 4 by 8 = 4/8 which DOES equal ½ so C is PROPORTIONAL

7.2 Scale Diagrams and Reductions Try example 2 on page 328 For part A you can use 2 ratios to set up a proportion and solve for the unknown measurement Think Linear Equations (yay) to solve for x we do the opposite operation and multiply both sides by 3.85m you get 0.077m, convert to cm you multiply by 100.

7.2 Scale Diagrams and Reductions Example 2 on Page 328 For part B you measure the width of the scaled truck with a ruler. Multiply the measurement by 50 because the actual truck is 50 times bigger. 50 x _____ = ______ 50 x 3.2cm = 160 cm

7.2 Scale Diagrams and Reductions Math practice homework: Page 329 – 331 Questions – 5, 6, 7, 9, 11, 13 Question 21 is for assessment, complete on a white sheet of paper. Due Friday May 6th. Remember – with reductions your scale factor will be less than 1! With question 21 what should you include? Title, border, name, date.

Q 21 – Mark Scheme >7 = 4 5-6 = 3 3-4 = 2 <2 = 1 Name Title/date Scale factor Diameter to specified scale Units given Conversion shown Included all 8 planets Blank paper Bonus – colour, paint, Pluto, organized, above and beyond!

7.2 – Sticky Math A rectangular playground has dimensions 24m by 16m. What are the dimensions of a scale diagram of this playground with a scale factor of 1/200? Consider appropriate units 12cm by 8cm

7.3 Similar Polygons What is a polygon (hint, these are all polygons)?

7.3 Similar Polygons Polygon – closed shape with straight sides Regular Polygon has equal sides and equal lengths. If you want to get technical – exactly 2 sides meet at a vertex.

7.3 Similar Polygons When one polygon is an enlargement or a reduction of another polygon, we say the polygons are similar. When 2 polygons are similar: Matching angles are equal AND Matching sides are proportional

7.3 Similar Polygons Example – are these polygons similar? Check matching angles (Q = U) Check matching sides (QR/UV = RS/VW = 1.5) All scale factors are equal so matching sides are proportional All angles are equal. These figures are similar.

7.3 Similar Polygons The following figures are similar Determine the length of JI and JF. Hint – what is the scale factor? HI/CD = 3.6/2.4 = 1.5 Multiply 1.2 by 1.5 and 1.8 x 1.5 JF = 1.8 cm and JI = 2.7 cm

7.3 Similar Polygons What would you do here? Find the length of ZY Hint – you will use skills gained on our last math unit! ST/WX = VU/ZY then 3/1.8 = 2/ZY Solve for ZY = 1.2 cm

7.3 Similar Polygons What if you need to measure angles. What do you need?

7.3 Similar Polygons What if you need to measure angles. What do you need? You do not need to bring a protractor to class, I will supply them, but you do need to know how to use one.

7.3 Similar Polygons Math Practice for HOMEWORK! Page 341 – 342 Questions 4, 5, 6, 9, 10, 11 and 16 Graph paper would be beneficial

7.4 Similar Triangles Two triangles are similar if they have the same shape but different size. In similar triangles Matching angles are equal Matching sides are proportional

7.4 Similar Triangles To write the similarity statement, corresponding angles and sides must match up Can you give 6 true statements from the similarity of these two triangles?

7.4 Similar Triangles When 2 polygons are similar: The measures of corresponding angles must be equal The ratios of the lengths of corresponding sides must be equal When 2 triangles are similar: AND OR

7.4 Similar Triangles X = 8 Y = 3

7.4 Similar Triangles Write a proportion that includes only 1 unknown. Solve. Remember to be consistent with your numerators and denominators. <P = 70, <Q = 80 and <R = 30 X = 8cm and y = 3cm

7.4 Similar Triangles Identify the 2 similar triangles and determine the missing sides. Use matching angles Write the similarity statement: ∆QPM ~ ∆NOM Match corresponding angles: ∠0 ∠P ∠M ∠M ∠N ∠Q

7.4 Similar Triangles Identify the 2 similar triangles and determine the missing sides. ∆QPM ~ ∆NOM Write a proportion that includes only 1 unknown X = 6m Y = 3.5m

7.4 Similar Triangles Identify the similar triangles and then solve for x and y. ∆ACE ~ ∆BCD Y = 9cm x = 15.75cm

7.4 Similar Triangles One triangle has two 500 angles. Another triangle has a 500 angle and an 800 angle. Could the triangles be similar? Explain. Remember angle sum theory….

7.4 Similar Triangles NOTE: With triangles all you need is to show that the three angles are congruent. In fact, knowing two angles are congruent means the third angle is also congruent.

7.4 Similar Triangles Practice questions for HOMEWORK Page 349 Questions 4, 5c, 6, 7, 9, 10, 11 and 15

7.4 Similar Triangles On a lined sheet of paper complete an enlargement and a reduction the triangles on page 343. Name, title, date 3 triangles – original, enlargement & reduction Use a protractor and a ruler Include all measurements, scale factors and units Organized Show your work Hand in by the end of class - TODAY

7.5 Reflections and Line Symmetry Symmetry No Symmetry Taj Mahal is a famous example of symmetry in architecture. Many parts of the building and grounds were designed and built to be perfectly symmetrical. Symmetry creates a sense of balance.

7.5 Reflections and Line Symmetry A figure is divided into 2 congruent parts using a line of symmetry (mirror image) One half of the figure is reflected exactly onto the other half A figure may have more than one line of symmetry Can be – VERTICAL, HORZINTAL or OBLIQUE

7.5 Reflections and Line Symmetry

7.5 Reflections and Line Symmetry

7.5 Reflections and Line Symmetry Can anymore lines of symmetry be drawn for a hexagon?

7.5 Reflections and Line Symmetry Investigate the lines of symmetry for these regular polygons Can you make a general statement describing the number of sides and the number of lines of symmetry in a regular polygon? Hint – yes you can…

7.5 Reflections and Line Symmetry The number of lines of symmetry is equal to the number of sides in a regular polygon. This only applies to regular polygons- irregular ones you have to wing it!

7.5 Reflections and Line Symmetry A line of symmetry is also called a line of reflection. If a mirror is placed along one side of a shape, the reflection image and the original shape together form one larger shape. The line of reflection is a line of symmetry of this larger shape.

7.5 Reflections and Line Symmetry Identify the lines of symmetry in each tessellation

7.5 Reflections and Line Symmetry Identify the lines of symmetry in each tessellation

7.5 Reflections and Line Symmetry Reflecting on the Cartesian Plane Give the coordinates for ABCD reflected on the X axis and y axis

7.5 Reflections and Line Symmetry

7.5 Reflections and Line Symmetry Identify the triangles that are related to the red triangle by a line of reflection. Describe the position of each line of symmetry (give coordinates).

7.5 Reflections and Line Symmetry

7.5 Reflections and Line Symmetry Math Practice for HOMEWORK! Page 358-359 Questions 3 (!), 5, 8, 9, 10

Energy Transformations You will need to have a labeled Science diagram for each station. Electric generators Electromagnetic Motor Oersted’s law

7.6 Rotations and Rotational Symmetry Rotational Symmetry – a figure has rotational symmetry if it can be turned around its centre to match itself in less than a 360o turn.

7.6 Rotations and Rotational Symmetry After one complete turn, the number of times that a figure matches itself is referred to as: Order of Rotational Symmetry OR Degree of Rotational Symmetry

7.6 Rotations and Rotational Symmetry Look at these REGULAR polygons. Determine the order of rotational symmetry for each.

7.6 Rotations and Rotational Symmetry Look at these REGULAR polygons. Determine the order of rotational symmetry for each. The degree or order of rotational symmetry is equal to the number of sides in a regular polygon.

7.6 Rotations and Rotational Symmetry Angle of Rotational Symmetry The minimum angle required for a shape to rotate and coincide with itself is:

7.6 Rotations and Rotational Symmetry Regular Polygon Summary

7.6 Rotations and Rotational Symmetry Regular Polygon Summary

7.6 Rotations and Rotational Symmetry Try these: What is the Order or Rotational Symmetry? What is the Angle of Rotational Symmetry? A – order is 2 angle is 180 B – order is 5 angle is 72 C – order is 4 angle is 90

7.6 Rotations and Rotational Symmetry What do you think the order of symmetry is for a circle? What if you know the angle of rotational symmetry and you are asked to find the order of rotation al symmetry? Infinite if you are given the angle you take 360 degrees and divide by that given angle to give you the order of symmetry. So if you are given an angle of 90 degrees than its order is 360 divided by 90 = 4

7.6 Rotations and Rotational Symmetry In grade 9 we create our own figures with rotational symmetry by rotating a shape around a vertex. Can you rotate pentagon ABCDE 90o clockwise around vertex E? Draw the rotation image.

7.6 Rotations and Rotational Symmetry Does your rotation image look like this? 90o clockwise around vertex E

7.6 Rotations and Rotational Symmetry Try this Rotate trapezoid FGHJ 120o counterclockwise about vertex F

7.6 Rotations and Rotational Symmetry Does your rotation image look like this? FGHJ 120o counterclockwise about vertex F

7.6 Rotations and Rotational Symmetry Copy this drawing (graph paper is your friend) Rotate rectangle ABCD: Draw and LABEL any rotations Look at the shape formed by the rectangle and all its images. Identify any rotational symmetry in this shape.

7.6 Rotations and Rotational Symmetry This new image has a rotational symmetry of 4.

7.7 – Identifying Types of Symmetry on the Cartesian Plane Translation is a term used in geometry to describe a function that moves an object a certain distance. The object is not altered in any other way. It is not rotated, reflected or re-sized.

7.7 – Identifying Types of Symmetry on the Cartesian Plane Draw rectangle ABCD and it’s transformation. Describe whether or not reflectional or rotational symmetry exist. Rotation of 180o about the origin

7.7 – Identifying Types of Symmetry on the Cartesian Plane The octagon that is formed has NO line of symmetry but has rotational symmetry about the origin. The octagon has an order of 2.

7.6 & 7.7 Math Practice Math Practice for HOMEWORK. You will want to complete this tonight. Page 366 – Questions – 4, 5, 6, 8, 9, 12 & 14 Page 373 – Questions – 3, 5, 6, 7 and 13 Assignment Due Thursday