Cambridge University Press  G K Powers Similarity and right-angled triangles Study guide 1

Cambridge University Press  G K Powers 2013 Similar figures and scale factors  Similar figures are the same shape and have the same proportions but are different sizes.  Corresponding (or matching) angles of similar figures are equal.  Corresponding sides of similar figures are in the same ratio.  Scale factor is the amount of enlargement or reduction. HSC Hint – Matching sides in similar figures are opposite angles that are equal in size. 2

Cambridge University Press  G K Powers 2013 Problems involving similar figures 1. Read the question and underline key terms. 2. Draw similar figures and label the information from the question. 3. Use a pronumeral (x) to represent an unknown side. 4. Write an equation using two fractions formed from matching sides. 5. Solve the equation. 6. Check that the answer is reasonable HSC Hint – Always check whether the answer makes sense. Write the answer in words. 3

Cambridge University Press  G K Powers 2013 Scale drawing Scale of a drawing = Drawing length : Actual length Scale is expressed in two ways: 1. Using units such as 1 cm to 1 m. 2. No units such as 1:100. HSC Hint – Scale drawing questions may require a measurement using a ruler. 4

Cambridge University Press  G K Powers 2013 Naming the sides of a right triangle The hypotenuse is opposite the right angle, the opposite side is opposite the angle θ and the adjacent is the remaining side. HSC Hint – Label the sides of a right triangle by starting with the hypotenuse. 5

Cambridge University Press  G K Powers 2013 Trigonometric ratios  The mnemonic ‘SOH CAH TOA’ is used to determine the trigonometric ratio. SOH: Sine-Opposite-Hypotenuse CAH: Cosine-Adjacent-Hypotenuse TOA: Tangent-Opposite-Adjacent  The order of the letters matches the ratio of the sides. HSC Hint – Learn the mnemonic and use it when there is an angle and sides in a right-angled triangle. 6

Cambridge University Press  G K Powers 2013 Finding an unknown side 1. Name the sides of the triangle. 2. Use the given side and unknown side x to determine the trigonometric ratio. Use SOH CAH TOA. 3. Rearrange the equation to make the x the subject. 4. Use the calculator to find x. Remember to check the calculator is set up for degrees. 5. Write the answer to the specified level of accuracy. HSC Hint – Always show the trigonometric equation with the known values. 7

Cambridge University Press  G K Powers 2013 Finding an unknown angle 1. Name the sides of the triangle. 2. Use the given sides and unknown angle θ to determine the trigonometric ratio. Use SOH CAH TOA. 3. Rearrange the equation to make θ the subject. 4. Use the calculator to find θ. Remember to check the calculator is set up for degrees. 5. Write the answer to the specified level of accuracy. HSC Hint – To find an angle on the calculator use the SHIFT key to select sin -1, cos -1 or tan -1. 8

Cambridge University Press  G K Powers 2013 Applications of right-angled triangles 1. Read the question and underline the key terms. 2. Draw a diagram and label the information. 3. Use trigonometry to calculate a solution. 4. Check that the answer is reasonable and units are correct. 5. Explain the answer in words and ensure the question has been answered. HSC Hint – In addition to trigonometry some questions in right triangles require the use of Pythagoras’ theorem. 9

Cambridge University Press  G K Powers 2013 Angles of elevation and depression  The angle of elevation is the angle measured upwards from the horizontal.  The angle of depression is the angle measure downwards from the horizontal. HSC Hint – Angle of elevation is equal to the angle of depression as they form alternate angles. 10