Section 8: Right Triangles

Transcription

1 Topic 1: The Pythagorean Theorem Topic 2: The onverse of the Pythagorean Theorem Topic 3: Proving Right Triangles ongruent Topic 4: Special Right Triangles: Topic 5: Special Right Triangles: Topic 6: Right Triangles Similarity Part Topic 7: Right Triangles Similarity Part Topic 8: Introduction to Trigonometry Part Topic 9: Introduction to Trigonometry Part Topic 10: ngles of Elevation and Depression Visit MathNation.com or search "Math Nation" in your phone or tablet's app store to watch the videos that go along with this workbook! 177

2 The following Mathematics Florida Standards will be covered in this section: G-O Explain how the criteria for triangle congruence (S, SS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. G-O Prove theorems about triangles; use theorems about triangles to solve problems. G-SRT Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G-SRT Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G-SRT Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G-SRT Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT Explain and use the relationship between the sine and cosine of complementary angles. G-SRT Use trigonometric ratios and Pythagorean Theorem to solve right triangles in applied problems. 178

3 Section 8 Topic 1 The Pythagorean Theorem onsider the following diagram and complete the two-column proof below. onsider the triangle below. cc D mm nn h aa 6 10 bb 8 What relationship exists between the length of the hypotenuse and the length of the legs?!"#$%&'!$! )*+,-./,0+%&%!20*304+ Pythagorean Theorem In a right triangle, the square of the hypotenuse (the side opposite to the right angle) is equal to the sum of the squares of the other two sides. Given: ~ ~ Prove: aa ) + bb ) = cc ) using triangle similarity. Statements Given Reasons orresponding sides of similar triangles are proportional Multiplication Property Of Equality 4. aa ) + bb ) = cccc + cccc Distributive Property Leg Hypotenuse aa cc 6. + DDDD = or mm + nn = cc aa ) + bb ) = cc ) 7. Leg bb aa ) + bb ) = cc ) 179

4 Let s Practice! 1. business building has several office spaces for rent. Each office is in the shape of a right triangle. If one side of the office is 11 feet long and the longest side is 15 feet long, what is the length of the other side? ET THE TEST! 1. baseball diamond is actually a square with sides of 90 feet. Part : If a runner tries to steal second base, how far must the catcher, who is at home plate, throw the ball to get the runner out? Try It! Part : Explain why runners try to steal second base more often than third base. 2. Mr. Roosevelt is leaning a ladder against the side of his son s tree house to repair the roof. The top of the ladder reaches the roof, which is 18 feet from the ground. The base of the ladder is 5 feet away from the tree. How long is the ladder? 180

5 Section 8 Topic 2 The onverse of the Pythagorean Theorem Suppose you are given a triangle and the lengths of the sides. How can you determine if the triangle is a right triangle?!"#$%&'!$! )*+,-./,0+%&%!20*304+ onverse Pythagorean Theorem If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. Pythagorean triple is a set of positive integers aa, bb and cc that satisfy the Pythagorean Theorem, aa ) + bb ) = cc ). The side lengths of a right triangle, 3, 4 and 5, form a Pythagorean triple. Prove that each set of numbers below is a Pythagorean triple. Let s Practice! 1. Zully is designing a bird feeder that her husband will build for the little birds that come to eat in the mornings. The bird feeder must be a right triangle. The first draft of her design is displayed to the right. Ø 5, 12, 13 Does this design contain a right triangle? Justify your answer. 8.5 Ø 8, 15, Ø 7, 24, Hypothesize if multiples of Pythagorean triples are still Pythagorean triples. Justify your answer. 181

6 Try It! 2. Mr. hris designed a Pratt Truss bridge with a structure that slanted towards the center of the bridge. In order to be a Pratt Truss bridge, the bridge has to contain right triangles in its design. However, his design was rejected by the construction firm. The firm said that Mr. hris s design failed to meet the Pratt Truss requirements. ET THE TEST! 1. lay designs roofs that form 2 congruent right triangles. His designs are flawless. He submitted his latest design to a firm along with three other contractors, and the firm selected lay s plan. Which of the following designs is lay s design? a. onsider the above representation of the bridge Mr. hris designed. Prove that the construction firm was correct in its rejection of Mr. hris s design b. What options does Mr. hris have to fix the design? Justify your answer. D

7 Section 8 Topic 3 Proving Right Triangles ongruent Let s review the four postulates that can be used to prove triangles are congruent. Let s Practice 1. onsider SSSSSS and RRRRRR in the diagram below. omplete the two column proof. P R Hypotenuse-Leg (HL) Theorem is another way to prove triangles are congruent.!"#$%&'!$! )*+,-./,0+%&%!20*304+ The Hypotenuse-Leg (HL) Theorem Two right triangles are said to be congruent if their corresponding hypotenuse and at least one of their legs are congruent. onsider the diagram below. S Given: SSSSSS and RRRRRR are right triangles and SSSS QQQQ. Prove: SSSSSS RRRRRR Q X Statements 1. SSSSSS and RRRRRR are right triangles. 1. Reasons Given Y Z Reflexive Property of ongruence List three statements that prove the triangles are congruent by the HL Theorem. 4. SSSSSS RRRRRR

8 2. onsider the following diagrams. Try It! xx xx + 3 3yy yy onsider and in the diagram below. omplete the two column proof. Find the values of xx and yy that prove the two triangles are congruent according to the HL Theorem. D Given: is perpendicular ; Prove: Statements 1. is perpendicular to. 1. Given Reasons 2. and are right angles Given

9 4. onsider the diagrams below. yy xx yy + 5 3xx + yy xx + 5 Find the values of xx and yy that make the right triangles congruent. ET THE TEST! 1. Engineers are designing a new bridge to cross the Intracoastal Waterway. elow is a diagram that represents a partial side view of the bridge. The bridge must be designed so that EEEEEE. Engineers have measured the support beams, represented by and EEEE in the diagram, and found they are both 120 ffff long. The engineers also determined that beams and EEEE are perpendicular to the bridge,. Point represents the midpoint of. E D omplete the two-column proof on the next page to prove EEEEEE. Statements Given Reasons 2. and EEEE and EEEEEE are right angles. 3. Definition of a midpoint EEEEEE

10 Section 8 Topic 4 Special Right Triangles: Use the Pythagorean Theorem to find the missing lengths of the following triangles. Let s Practice! 1. onsider the following triangle. Prove that the ratio of the hypotenuse to one of the legs is 2: Find the hypotenuse of a triangle with legs equal to 5 cm. hoose three patterns that you observe in the three right triangles and list them below. 186

11 Try It! 3. Find the length of the sides of a square with a diagonal of 25 ) _ meters. ET THE TEST! 1. onsider the drawing below. II The Tilley household wants to build a patio deck in the shape of a triangle in a nice corner section of their backyard. They have enough room for a triangular deck with a leg measuring 36 feet. What will the length of the longest side be? RR 43 TT GG Part : What is the perimeter of the figure? Part : Write a 3 sentence long short story about the drawing and the calculations made in Part. 187

12 Section 8 Topic 5 Special Right Triangles: Use the Pythagorean Theorem to find the missing lengths of the following triangles. Let s Practice! 1. The length of a hypotenuse of a right triangle is 17 yards. Find the other two lengths Try It! 2. right triangle has a leg with a length of 34 and a hypotenuse with a length of 68. student notices that the hypotenuse is twice the length of the given leg and says that this means it is a triangle. If the student is correct, what should the length of the remaining leg be? Explain your answer. onfirm your answer using the Pythagorean Theorem. 3 hoose three patterns that you observe in the three right triangles and list them below. 188

13 ET THE TEST! 1. The base of the engineering building at Lenovo Tech Industries is approximately a triangle with a hypotenuse of about 294 feet. The base of the engineering building at sus Tech Industries is approximately an isosceles right triangle with a side about feet. Section 8 Topic 6 Right Triangles Similarity Part 1 Make observations about the following triangles. What is the difference between the perimeters of the two buildings? Round your answer to the nearest hundredth. F D E These triangles are similar by the. onsider the diagrams below. D Make observations about and. 189

14 !"#$%&'!$! )*+,-./,0+%&%!20*304+ Right Triangle ltitude Theorem If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Try It! 2. roof s cross section forms a right angle. onsider the diagram below that shows the approximate dimensions of this cross section. Let s Practice! R 1. onsider the following diagram. 8.9 m h 4.1 m 13 m D h 5 m S 7.9 m F O 12 m a. Identify the similar triangles in the above diagram. a. Identify the similar triangles represented in the above figure. b. Find h in the above diagram. b. Find the height h of the roof represented above. 190

15 !"#$%&'!$! )*+,-./,0+%&%!20*304+ Geometric Mean Theorem: ltitude Rule In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments. onsider the following diagram. cc nn D h mm aa = DDDD bb D an we accept ~ as a given statement? Justify your answer.!"#$%&'!$! )*+,-./,0+%&%!20*304+ Geometric Mean Theorem: Leg Rule In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. omplete the following two-column proof to prove that h = mmmm. Statements Given Reasons = DDDD 2. i j = j k 2. or Multiplication Property of Equality D = 4. h = mmmm

16 Section 8 Topic 7 Right Triangles Similarity Part 2 3. onsider the diagram below. Let s Practice! 1. onsider the diagram below and find the value of xx. xx yy DD 25 mm xx zz mm 2. onsider the diagram below and find the value of yy. Given: ~ 2 Prove: xx = 20 mm yy = 16 mm zz = 9 mm y 5 192

17 Try It! 4. onsider the diagram below. 3 xx yy 4 5. cruise port, a business park, and a federally protected forest are located at the vertices of a right triangle formed by three highways. The port and business park are 6.0 miles apart. The distance between the port and the forest is 3.6 miles, and the distance between the business park and the forest is 4.8 miles. service road will be constructed from the main entrance of the forest to the highway that connects the port and business park. What is the shortest possible length for the service road? Round your answer to the nearest tenth. z Find the values of xx, yy, and zz to the nearest tenth. 193

18 ET THE TEST! 1. onsider the statement below. In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments. Which of the following figures is a counterexample of the statement above? 2. shopping center has the shape of a right triangle with sides measuring meters, 600 meters, and 1,200 meters. During the holidays and busy seasons, the shopping center is so crowded that it needs another walkway. The owners will construct the walkway from the right angle to the hypotenuse. They want to use the shortest possible length for the walkway. a. Determine the length of the segment of the hypotenuse adjacent to the shorter leg. meters. b. Determine the length of the new walkway meters D

19 Section 8 Topic 8 Introduction to Trigonometry Part 1 Let s examine the three main trigonometric ratios. omplete the statements below. In previous lessons, we learned that the lengths of the sides of a right triangle have a certain relationship, which allows us to use the. onsider the following right triangles. = = = x} tsstuvwx wt wjx ~k} x jzstwxk{ux x} ~Ä~Åxkw wt wjx ~k} x jzstwxk{ux x} tsstuvwx wt wjx ~k} x x} ~Ä~Åxkw wt wjx ~k} x Let s Practice! 1. onsider the figure below. For angle, find the ratio of the opposite leg to the hypotenuse rsstuvwx yzstwxk{ux = Find the same ratio for angle. = 36 T The ratio of the lengths of any 2 sides of a right triangle is a. Find the sine, cosine, and tangent of TT for the figure. 195

20 Try It! Now, let s consider the figure below. 2. onsider the figure below a. Find sin for the above triangle. b. Find cos for the above triangle. The triangle above is a special right triangle known as the triangle. We know that the two non-right angles measure. Write proportions for sin, cos, and tan of the acute angles of the triangle. Use a calculator to verify the proportions. c. What do notice about the values of sin and cos? If there is an unknown length, we can set up an equation to find it. 196

21 Let s Practice! 3. onsider the following figure. xx yy Try It! 4. onsider the figure below. 62 yy 18 a. Which trigonometric function should you use to find the value of xx? 28 b. Write an equation to find xx in the above figure. Determine the value of yy. c. Find the value of xx in the above figure. 197

22 Section 8 Topic 9 Introduction to Trigonometry Part 2 Given the lengths of sides, we can use trig functions to find missing angles by using their inverses: sin äã, cos äã, and tan äã. Try It! 2. onsider the triangle below. U 40 M Let s Practice! 1. onsider the triangle below D Find tanmm, cosdd, mm DD and sinmm for the triangle. Find cos, sin, mm and mm for the triangle. 198

23 ET THE TEST! 2. onsider the triangle below. 1. The picture below shows the path that Puppy Liz is running. The electrical post is 40 feet tall. Puppy Liz usually starts at the bench post and runs until she gets to the fire hydrant, rests, and then she runs back to the bench. How far does Puppy Liz run to get to the fire hydrant? xx yy 40 ft Which of the following measurements represents the perimeter and area of the triangle above? Puppy Liz runs feet. Perimeter: units rea: square units Perimeter: units rea: square units Perimeter: units rea: square units D Perimeter: units rea: square units 199

24 3. Yandel will place a ramp over a set of stairs at the backyard entrance so that one end is 5 feet off the ground. The other end is at a point that is a horizontal distance of 40 feet away, as shown in the diagram. The angle of elevation of the ramp is represented by ee. Section 8 Topic 10 ngles of Elevation and Depression ngles of elevation are angles the horizon. ngles of depression are angles the horizon. 5 ft ramp In your own words, explain what are angles of elevation and angles of depression. 40 ft 40#$%. ( Explain how the properties of a right triangle fit into the properties of angles of elevation or depression. What is the angle of elevation to the nearest tenth of a degree? Describe a situation where you would deal with angles of elevation. Describe a situation where you would deal with angles of depression. 200

25 If your eye level is 6 feet above the ground, then what is the vertical distance from your eyes to the birds? How can you use this information to find your horizontal distance from the birds? Let s Practice! 1. Suppose that an airplane is currently flying at an altitude of 39,000 feet and will be landing on a tarmac 128 miles away. Find the average angle at which the airplane must descend for landing. Round your answer to the nearest tenth of a unit. 2. onsider the diagram below that represents someone s eye level as he looks at his dog. Find the value of xx, and round to the nearest hundredth of a foot. 25 xx 13 ft 201

26 Try It! 3. If Lionel has an eye level of 5 feet above the ground and he is standing 40 feet from a flagpole that is 32 feet tall, then what is the angle of elevation? ET THE TEST! 1. man is 6 feet 3 inches tall. The tip of his shadow touches a fire hydrant that is 13 feet 6 inches away. What is the angle of elevation from the base of the fire hydrant to the top of the man s head? Round to the nearest tenth of a degree D Suppose that you are standing on a hill that is 59.5 ft tall looking down on a lake at an angle of depression of 48. How far are you from the lake? Round your answer to the nearest foot. Test Yourself! Practice Tool Great job! You have reached the end of this section. Now it s time to try the Test Yourself! Practice Tool, where you can practice all the skills and concepts you learned in this section. Log in to Math Nation and try out the Test Yourself! Practice Tool so you can see how well you know these topics! 202