Congruence and Similarity in Triangles Pg. 378 # 1, 4 8, 12. Solving Similar Triangle Problems Pg. 386 # 2-12

Transcription

1 UNIT 7 SIMILAR TRIANGLES AND TRIGONOMETRY Date Lesson TOPIC Homework May May Congruence and Similarity in Triangles Pg. 378 # 1, 4 8, 12 Solving Similar Triangle Problems Pg. 386 # 2-12 May Exploring Similar Right Triangles Pg. 393 # 1-4 May10 OPT Mid- Chapter Review Pg. 390 # 1-10 May The Primary Trigonometric Ratios Pg. 398 # 2, 3, 5, 7-13 May Solving Right Triangles Pg. 403 # 1 4, 7, 8a, 9 11, 13ac, 14 May Solving Right Triangle Problems Pg. 412 # 1 6, 10, 12, 14 May Solving Right Triangle Problems Two-step Problems Pg. 413 # 11, 13, 15 17, 20 May 17/ Review for Unit 7 Test Pg. 416 # 1 16 Plus Review WS 7.8 May TEST- UNIT 7

2 MPM 2D Lesson 7.1 Congruence and Similarity in Triangles A D B C E F P X Y Z Q R Using a ruler and protractor measure each of the following very carefully. Measure sides to the nearest mm and angles to the nearest degree. AB = A = DE = D = AC = B = DF = E = BC = C = EF = F =

3 PQ = P = XY = X = PR = Q = XZ = Y = QR= R = YZ = Z = When comparing ABC and PQR, what do you notice about the lengths of the sides and the measure of the angles? For ABC and DEF, AB corresponds to side in DEF. A corresponds to in DEF. AC corresponds to side in DEF. B corresponds to in DEF. BC corresponds to side in DEF. C corresponds to in DEF. For ABC and DEF, complete the following. AB DE A = D = AC DF B = E = BC EF C = F = What do you notice about the ratios of the corresponding sides? What do you notices about the corresponding angles? Triangles when the ratios of the lengths of the corresponding sides are and the measures of the corresponding angles are the triangles are called The scale factor or scale ratio is the measure of the amount of enlargement or reduction from one similar triangle to the other. The scale factor is the ratio of any 2 corresponding sides of similar triangles.

4 Properties of Similar and Congruent Triangles If ABC XYZ and the scale factor is AB n then: XY the length of any side or altitude of ABC = n(length of corresponding side or altitude of XYZ) the perimeter of ABC = n(perimeter of XYZ) the area of ABC = n 2 (area of XYZ) To prove that 2 triangles are similar. Angle-Angle Similarity (AA) - Two corresponding pairs of angles share the same measure. Side-Side -Side Similarity (SSS) -- Three corresponding pairs of sides have a common ratio Side-Angle -Side Similarity (SSS) -- Two corresponding pairs of sides have a common ratio and the contained angles share the same measure. To prove that 2 triangles are congruent. Side-Side -Side Congruity (SSS) Two triangles are congruent if all three sides have equal measures. Side-Angle -Side Congruity (SAS) Two triangles are congruent if two sides and the contained angle have the same measures. Angle-Side-Angle Congruity (ASA) - Two triangles are congruent if two angles and the contained side have the same measures. Hypotenuse-Side (HS) - Two triangles are congruent if their hypotenuses and one of the other sides have the same measure. Hypotenuse-Angle (HA) - Two right triangles are congruent if their hypotenuses and one of the acute angles have the same measure Finally, if two triangles are congruent they are also similar, but two similar triangles are not necessarily congruent. Ex. 1 Explain why one of the triangle below is similar to ABC and the other is not. X A P B 3 C Y 6 Z Q 5.3 R

5 Ex. 2 Prove the following triangles congruent and determine the value of each lower-case letter. a) G 8 m J x 140 H a I Ex. 3 Show that triangle QYN is congruent to triangle QYP. Pg. 378 # 1, 4 8, 12

6 MPM 2D Lesson 7.2 Solving Similar Triangle Problems Ex. 1 How tall is the tree below? 2 m 3 m 11 m Ex. 2 Jay stands on level ground and looks at the mirror on the ground that is 2 m from his feet. He can see the top of a flag pole that is 7 m from the mirror. If his eyes are 1.72 m from the ground, how tall is the flag pole? Pg. 386 # 2-12

7 MPM 2D Lesson 7.3 Exploring Similar Right Triangles 1. Use the triangles below to complete the tables on the next page. G E C A B D F 2. Use the triangles below to complete the tables on the next page. W X T P Q R S

8 Complete the tables below for each of the given similar triangles. 1- Give all answers correct to 3 decimal places. Triangle sin opposite hypotenuse cos adjacent hypotenuse opposite tan adjacent ABC BC AC AB AC BC AB ADE DE AE AD AE DE AD AFG FG AG AF AG FG AF sin cos tan 2- Give all answers correct to 3 decimal places. Triangle sin opposite hypotenuse cos adjacent hypotenuse opposite tan adjacent PQT QT PT PQ PT QT PQ PRX RX PX PR PX RX PR PSW SW PW PS PW SW PS sin cos tan The Primary Trigonometric Ratios can only be used for right triangles. Trig ratios are simply the ratio of the sides of a right angled triangle. Each trig ratio represents the ratio of two different sides.

9 For the triangle below and, For the triangle below and, O p p o hypotenuse hypotenuse adjacent s i t e adjacent opposite The side that is labelled opposite and the side labelled adjacent depends on which angle is being used. Unless told otherwise always find the length of a side to 1 decimal place and the measure of an angle to the nearest degree. SINE : COSINE : TANGENT sin cos : opposite hypotenuse tan adjacent hypotenuse opposite adjacent SOH CAH TOA Ex. 1 Find the value of x, correct to 1 decimal place. A Ex. 2 Find A to the nearest degree. A x B 32 C B C

10 Ex. 3 Solve each of the following triangles. (ie: find all missing sides and angles) a) P b) 4 Y 46 Z 12 X Q 5 R Pg. 393 # 1-4

11 MPM 2D Lesson 7.4 The Primary Trigonometric Ratios Ex. 1 Determine length x in each triangle. Round your answer to one decimal place. a) 8.4 cm 15 x b) 8.2 km x 42 Ex. 2 Determine the measure of A, to the nearest degree. A B 12.1 cm 8.2 cm C

12 Ex. 3 A hot-air balloon on the end of a taut 95 m rope rises from its platform. Sam, who is in the basket, estimates that the angle of depression to the rope is about 50. a) How far, to the nearest metre, did the balloon drift horizontally? b) How high, to the nearest metre, is the balloon above ground? c) Viewed from the platform, what is the angle of elevation, to nearest degree, Ex. 4 A wheelchair ramp is safe to use if it has a minimum angle of 4.8 and a maximum angleof What are the minimum and maximum slopes of such a ramp? Round your answers to 2 decimal places. Pg. 398 # 2, 3, 5, 7-13

13 MPM 2D Lesson 7.5 Solving Right Triangles When you are asked to solve a triangle, you are being asked to find all of the unknown angle measures and side lengths. Ex. 1 Solve the following triangles. a) b) Ex. 2 During its approach to Earth, the space shuttle s glide angle changes. When the shuttle s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. a) What is its glide angle? Round your answer to the nearest tenth of a degree.

14 b) When the space shuttle is 5 miles from the runway, its glide angle is about 19. Find the shuttle s altitude at this point in its descent. Round your answer to the nearest tenth. Ex. 3 During a flight, a hot air balloon is observed by two persons standing at points A and B as illustrated in the diagram. The angle of elevation of point A is 28. Point A is 1.8 miles from the balloon as measured along the ground. Round answers to the nearest tenth. a) What is the height, h, of the balloon? h B A b) Point B is 2.4 miles from point A. Find the angle of elevation of point B. Pg. 403 # 1 4, 7, 8a, 9 11, 13ac, 14

15 MPM 2D Lesson 7.6 Solving Right Triangle Problems Ex. 1 A carpenter leans a 4.3 m ladder up against a wall. If it reaches 3.8 m up the wall, determine, to the nearest degree, the angle the ladder makes with the wall. Ex. 2 A missile is launched at an angle of elevation of 80. If it travels in a straight line, what is its altitude, correct to 1 decimal place, when it hits the training drone 15 km down range?

16 Ex. 3 Catalina s parents have a house with a triangular front lawn as shown. They want to cover the lawn with sod. How much would it cost to put sod in, if it costs $13.75 per square metre? Pg. 412 # 1 6, 10, 12, 14

17 MPM 2D Lesson 7.7 Solving Right Triangle Problems Two-Step Problems Ex. 1 Jon is standing on a 40 m high seaside cliff flying a kite. The angle of depression of the kite string is 38. If the kite string is m long, how far above the water is the kite? Ex. 2 Find the value of x m x

18 Ex. 3 From the bridge of The Maid of the Mist on the Niagara River, the angle of elevation to the top of Niagara Falls is 64. The angle of depression to the bottom of the falls is 6. If the bridge of the boat is 2.8 m above the water, calculate the height of the falls, correct to one decimal place. Ex. 4 Find the value of h, correct to one decimal place. x 50.0 m y h Pg. 413 # 11, 13, 15 17, 20