Measurement (MM3) Similarity of Two- Dimensional Figures & Right- Angled Triangles. Name... G. Georgiou

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1 Measurement (MM3) Similarity of Two- Dimensional Figures & Right- Angled Triangles Name... G. Georgiou 1 General Mathematics (Preliminary Course)

2 Calculate Measurements from Scale Diagrams Solve Practical Problems using Scale Diagrams and Factors, Similarity and Trigonometry The scale factor of an object tells us how much bigger or smaller a drawing is compared to its real life counterpart. Scale factors are often presented as fractions but can also be presented as ratios. In these cases, the first number in the ratio represents the measurement for the drawing and the second number represents the measurement for the real life object. When writing a scale factor in ratio form, the numbers MUST be written in the same units. For example - 1:4 means 1cm on the drawing represents 4cm in real life. Example 1 A surveyor makes a scale drawing of a block of land and uses a scale of 10 millimetres to 5 metres. (a) Write the scale as a simplified ratio. (b) One side of the block is 120 m long. What length represents this side on the scale drawing? (c) If another side is 8 mm long on the drawing, what is its actual length? 2 General Mathematics (Preliminary Course)

3 H.S.C. Question (2) 3 General Mathematics (Preliminary Course)

4 H.S.C. Question (3)... 4 General Mathematics (Preliminary Course)

5 H.S.C. Question (4) The map shows the Namoi catchment region in NSW. 2 Walgett N N amoi River Narrabri Keepit Dam Gunnedah Kilometres Tamworth The land use within the shaded area is mainly forestry and conservation. Using the scale given, calculate this area, to the nearest square kilometre. RAFT Activity Ex 3.03 Q 2-6, 8, 11 5 General Mathematics (Preliminary Course)

6 Calculate Scale Factors for Similar Figures Use Scale Factor to Solve Problems involving Similar Figures Consider the following two similar figures. If the first figure is the original shape and the second is the scale drawing, then it is quite clear that the original has simply been doubled. This means that these shapes have a scale factor of 2. To determine this value, we can use the following formula. Formula To find the scale factor of 2 objects or images: Scale Factor = NOT PROVIDED ON HSC FORMULA SHEET Example 5 Figures A and B are similar. Determine the scale factor given B is the image. 6 General Mathematics (Preliminary Course)

7 Example 6 The two figures below are similar. Determine the enlargement factor, and hence calculate the value of p. Example 7 A 4m flagpole casts a shadow 7m long. At the same time a nearby multistorey building casts a shadow of 48m. How tall is the building? 7 General Mathematics (Preliminary Course)

8 Example 8 Calculate the value of the pronumeral, given that the sides labelled 27 and 36 are corresponding sides. Example 9 Calculate the value of h. 8 General Mathematics (Preliminary Course)

9 H.S.C. Question (10) 9 General Mathematics (Preliminary Course)

10 H.S.C. Question (11) 10 General Mathematics (Preliminary Course)

11 In the past, questions about similarity have also involved area and volume, not just length. In these cases: ~ Area scale factor will be the length scale factor squared ~ Volume scale factor will be the length scale factor cubed Example 12 The two figures below are similar. 6 cm 20 cm (a) Write the enlargement factor in simplest form. (b) If the area of the smaller trapezium is 20 cm 2, determine the area of the larger trapezium. 11 General Mathematics (Preliminary Course)

12 Example 13 Activity Ex 3.02 Q 1-3, 5, 7, 8, General Mathematics (Preliminary Course)

13 Recognise that the Ratio of Matching Sides in Similar Right- Angled Triangles is Constant for Equal Angles Measure the sides of these triangles. Record your results in the table below. Smaller Triangle Larger Triangle Larger Triangle Side Smaller Triangle Side Smallest Side Largest Side Remaining Side What do you notice about the values in the last column? What does this tell us about triangles with the same angles? This rule applies to all triangles with equal angles, including right- angled triangles. This is where trigonometry comes from; it is the branch of mathematics dealing with the relationship between the sides and angles of triangles. 13 General Mathematics (Preliminary Course)

14 Calculate Sine, Cosine and Tangent Ratios Trigonometry helps us to calculate lengths and angles that are difficult or impossible to measure in real life. This booklet focuses on right- angled triangle trigonometry. Skill 1: Naming the Sides of a Right- Angled Triangle 1. Hypotenuse: the longest side of the triangle opposite the right angle. 2. Opposite: the side directly opposite the angle in question. 3. Adjacent: the remaining side (or the side next to the angle in question). Example 14 Examine the right- angled triangle below. A α B Φ C (a) Identify the hypotenuse, opposite and adjacent sides of ΔABC in relation to α. (b) Identify the hypotenuse, opposite and adjacent sides of ΔABC in relation to φ. 14 General Mathematics (Preliminary Course)

15 Skill 2: Defining Sine, Cosine and Tangent Ratios All right- angled triangles with the same angles have their sides in the same ratio. Here are 3 ratios that will help us use this knowledge to our advantage. Formula sin θ = cos θ = tan θ = NOT PROVIDED ON HSC FORMULA SHEET Example 15 Determine the value of sin θ, cos θ, and tan θ for the following right- angled triangles. (a) (b) Example 16 If sin θ =!, determine the values of cos θ and tan θ. " 15 General Mathematics (Preliminary Course)

16 Example 17 If tan θ =!" #, determine the values of sin θ and cos θ for the right- angled triangle below. 36 Θ Think!!! Why is it that the value of sin θ and cos θ are between - 1 and 1? Why can tan θ have a value greater than 1 or negative 1? 16 General Mathematics (Preliminary Course)

17 Skill 3: Defining a Degree as 60 Minutes and a Minute as 60 Seconds Ø 1 = 60 minutes (60 ) Ø 1 = 60 seconds (60 ) This means angles can be written in two equivalent forms: as decimals and in DMS (degrees, minutes, seconds) form = since 30 minutes is half a degree You can enter any decimal or DMS angle into your calculator and convert between the two forms by pressing your DMS button, which leads us to Skill 4: Converting Angles from Decimal to DMS Form To convert an angle in decimal form to DMS form, complete these steps: 1. Enter the angle and press = (enter 22.74=) 2. Press the DMS button ( ) 3. Your calculator should read Press the DMS button again to change the angle back to a decimal. Example 18 Write these angles in degrees, minutes and seconds form. (a) (b) Skill 5: Entering Angles in DMS Form into the Calculator If asked to enter an angle in DMS form into your calculator, you would follow these steps: 1. Type 22 ( ) ( ) ( ) 4. Press = (this enters the angle ) 17 General Mathematics (Preliminary Course)

18 Skill 6: Rounding Answers in DMS Form When rounding angles in DMS form, minutes and seconds only go up to 60, so the halfway mark to decide to round up or down to is 30. Example 19 Round the following angles correct to the nearest degree. (a) (b) (c) Example 20 Round the following angles correct to the nearest minute. (a) (b) (c) Skill 7: Solving Trigonometric Equations To find an angle in a trigonometric equation, just press SHIFT and enter everything as else as given. Example 21 Solve the following trigonometric equations and round your answers correct to the nearest minute where applicable. (a) sin θ =! "#... (b) cos θ = (c) tan θ=! " General Mathematics (Preliminary Course)

19 Activity Complete this sheet. 19 General Mathematics (Preliminary Course)

20 Use Trigonometric Ratios to find an Unknown Side- Length in a Right- Angled Triangle, when the Unknown Side- Length is in the Numerator of the Ratio to be Used NOTE: we will cover questions with the pronumeral in the denominator but they will NOT be formally assessed. The 3 trigonometric ratios of sin, cos and tan can be used to find the value of unknown sides- lengths in right- angled triangles. Just substitute the information that you are given in the question and use your equation solving skills. Example 22 Calculate the value of the pronumeral to 3 significant figures. (a) (b) (c) (d) 20 General Mathematics (Preliminary Course)

21 Example 23 An 8m flagpole has two support wires attached 5 metres from the top. These wires make an angle of 57 with the ground. Calculate the length of each wire correct to 1 decimal place. Activity Ex 3.06 Q General Mathematics (Preliminary Course)

22 Use Trigonometric Ratios to Find the Size of an Unknown Angle in a Right- Angled Triangle, correct to the Nearest Degree The 3 trigonometric ratios of sin, cos and tan can also be used to find the value of unknown angles in right- angled triangles. Just substitute the information that you are given in the question and press SHIFT when entering all the information into the calculator. Ensure you correctly round your angle correct to the nearest degree. Example 24 Calculate the value of θ, correct to the nearest degree. (a) (b) Example 25 Δ XYZ has Y= 90, XY = 12.2cm and YZ = cm. Calculate the value of Z correct to the nearest degree. 22 General Mathematics (Preliminary Course)

23 Example 26 A wheelchair access ramp rises 1m for every 4.5m horizontally. If government policy states that all wheelchair access ramps must incline no greater than 5, will this wheelchair ramp pass in accordance with government policy? Justify your answer with calculations Activity Ex 3.07 Q General Mathematics (Preliminary Course)

24 Solution of Problems involving Angles of Elevation and Depression, given the Appropriate Diagram Determine whether an Answer seems Reasonable by Considering Proportions within the Triangle Solve Practical Problems using Scale Diagrams and Factors, Similarity and Trigonometry Angles in the real world can be calculated when looking up an object or looking down at an object. These are called angles of elevation and angles of depression. Example 27 From a point 3.4 kilometres from the foot of a cliff, the angle of elevation to the top of the cliff is 17. Find the height of the cliff, correct to one decimal place. 24 General Mathematics (Preliminary Course)

25 Example 28 A plane is at an altitude of 2500 metres and descending towards the runway that is a horizontal distance of 7700 metres away. Find the angle of depression of the plane s path (to the nearest degree). Example 29 From the top of a 115 metre building, the angle of depression to a water fountain is 22. Calculate the distance from the foot of the building to the water fountain. 25 General Mathematics (Preliminary Course)

26 Example General Mathematics (Preliminary Course)

27 Example 31 Activity Ex 3.10 EVENS 27 General Mathematics (Preliminary Course)