4.1 Converse of the Pyth TH and Special Right Triangles

Transcription

1 Name Per 4.1 Converse of the Pyth TH and Special Right Triangles CONVERSE OF THE PYTHGOREN THEOREM Can be used to check if a figure is a right triangle. If triangle., then BC is a Eample 1: Tell whether the given side lengths would form a right triangle. a) 29, 24, 16 b) 6,, 9 pplications of the Converse (2 LEGS) (HYP) If a 2 + b 2 = c 2, then BC is a triangle. If a 2 + b 2 < c 2, then BC is. (shorter legs) If a 2 + b 2 > c 2, then BC is. (longer legs) LONGER LEGS cute super model SHORTER LEGS obese (obtuse) Eample 2: Classify the triangles as right, acute, or obtuse. a) 17, 4, 14 b) 5, 4, c) 16,, 8 USES OF THE PYTHGOREN THEOREM When you already know it s a RIGHT triangle Given 3 sides lengths of triangle Given 2 side lengths find the 3 rd side length Plug into Pyth. Thrm find out if it s right, acute, or obtuse The Flip If Neat things you can do with proportions (will come in handy later ), then If The Swap, then 1

2 TRINGLE THEOREM In a triangle, the etended ratio of the sides is 1: 1: * Note: Legs are always b 45 a c : : E#1: Find the value of the variables. Leave answers as simplified radicals. 1. Solve for. 2. If z =, then find the value of. 3. What is the value of in the triangle? 45 Is this a ? Why? TRINGLE THEOREM In a triangle, the etended a ratio of the sides is 1: : 2 6 b c : 3 : E#2: Find the indicated length. Leave answers as simplified radicals. 1. Solve for. 2. If c =, find the values of a 3. Solve for and. and b. c 15 2

3 Eample 3: In groups of two find the missing sides indicated in each problem. If possible, leave answers in simplified radicals. 1. If, then find the value of. y. 11 B. 11 C. 22 D. 11 z 30º 2. Which would serve as a countereample to the statement below? ny three side lengths of a triangle forms a right triangle.. 2, 9, 5 B. 3, 4, 5 C. 21, 20, 19 D. 11, 61, 60 3) Find the value of n and m. 4) Find the value of k. n m 18 k *Problems that are easily confused What to Use? What to Use? 1. In the figure below, n is a whole number. Find the value of n. Eplain. 2. In the figure below, n is a whole number. What is the smallest possible value of n? Eplain. 3. In the figure below, n is a whole number. What is the largest possible value of n? Eplain. n. 4 n k n k n n n B. 8 B. 7 B. 10 C. 16 C. 8 C. 11 D. 2 D. 9 D. 12 3

4 4.2 More Special Right Triangles RECTNGLE SQURE EQUILTERL TRINGLE diagonal divides a rectangle into two s. Use Pythagorean Theorem diagonal divides square into two s. Use ratios The altitude (height) divides it into two s. Use ratios E#3: If necessary in the problems below, leave your answer as a simplified radical. 1. YELP is a square and its diagonal has a length 18 inches. Find the length of one of its sides. 2. ΔHT is an equilateral triangle, find the length of its altitude. Y E 10 P L H T b. Find the area of YELP. b. Find the area of ΔHT. 3. If the length of the diagonal of a rectangle is 39 inches and the width of the rectangle is 15 inches, then find the area of the rectangle. (Draw a st inkin picture!) 4. Find length of the diagonal of a square with whose perimeter is 32 feet. (Draw a stinkin picture!) 5. Find the side length of an equilateral triangle whose height 21meters. (Draw a stinkin picture!) 6. You are standing 18 feet from a building. The vertical distance from the ground to your eye is 5.5 feet. Determine the height of the building. Round your answer to the nearest tenth. 30 4

5 REVIEW 1. Classify a triangle with side lengths 14, 19, and.. right B. acute C. obtuse D. not a triangle 2. If j = 6 in the right triangle below, then find the value of k.. 6 B. 6 C. 12 D. 12 j k l 3. Find the perimeter of an equilateral triangle whose altitude (height) is feet.. 9 B. 18 C. 54 D Solve for and eplain.. 73 B. 53 C. 56 D SPIRL REVIEW - Transformations THE RULES Preimage (1, 3) REFLECTION in the -ais Preimage (1, 3) ROTTION 90, REFLECTION in the y-ais ROTTION (1,3) 1. The point (4, -2) is transformed to (2, 4). Name the transformation. (1,3) 2. The point B(4, -2) is transformed to B (4, 2). Name the transformation. a. Reflection across the -ais b. Reflection across the y-ais c. 90 rotation clockwise about the origin d. 90 rotation counterclockwise about the origin a. Reflection across the -ais b. Reflection across the y-ais c. 90 rotation clockwise about the origin d. 180 rotation about the origin 3. The point H(1, 5) is transformed to H (-5, 1). Name the transformation. 4. The point G(8, -9) is transformed to G (-8, 9). Name the transformation. a. Reflection across the -ais b. Reflection across the y-ais c. 90 rotation clockwise about the origin d. 90 rotation counterclockwise about the origin a. Reflection across the -ais b. Reflection across the y-ais c. 90 rotation clockwise about the origin d. 180 rotation about the origin 5

6 SPIRL REVIEW - rea and Perimeter Perimeter rea of a rea of a REFRESH your Rectangle Triangle memory! rea of a Circle Circumference 1. Find the perimeter of the figure below. (ll line segments meet at right angles) Find the area of the triangle below. 7 9 Lesson Preview 1) Label the hypotenuse H 2) Put a smiley face on the side adjacent to 3) Put a star on the side opposite of Question: Would your answer change if you labeled the side opposite of and adjacent to? Why or why not? Practice: Label an O for opposite side, for adjacent side, and an H for hypotenuse based on the given angle. 1. Label from 2. Label from 3. Label from I R M U R G I K G *Which side is always labeled the same regardless of what angle you start from? O,, or H? Why? 6

7 4.3 Intro to Trigonometric Ratio Identifying the Hypotenuse, opposite, and adjacent side of a right Reference angle Hypotenuse Opposite djacent 1. Using J as your reference angle, label the hypotenuse, opposite, and adjacent sides. J K L 2. Would your answers change if your reference angle was L? Why or why not? 3. Can you use the right angle as your reference angle? Why or why not? How to set it up! The sine, cosine, and tangent of DEF E from reference angle D F D *Each trigonometric ratio depends on which acute angle you are starting from Here s How to MEMORIZE the TRIG RTIOS SOH CH TO sin = cos = tan = TYPE 1: Given side lengths of a right triangle, find the sine, cosine, and tangent 1. S a. b c. d. the same as T 40 R 2. Find the trigonometric ratios of each right triangle. B C Find the trigonometric ratios of each right triangle. J 16 K 8 L 7

8 4. Find the sin if cos =. 5. Find the cosg and sing if tang = 3. P I C B G TYPE 2: Given 1 SIDE and 1 CUTE NGLE, find a side length Steps to solve: 1) Pick a given CUTE angle to start from (Mark it up and label OPP, DJ, HYP) 2) Use SOH-CH-TO (pick 2 sides for ratio -1 you WNT & -1 you HVE) 3) Cross multiply and solve! *Don t round until the end! Find the missing side indicated. Round any values to the tenths place

9 4.4 pplying Trigonometric Ratios USING TRIGONOMETRIC RTIOS 1. Mark your reference angle 2. Label the Hypotenuse, the Opposite and djacent side (*based on given acute angle) 3. Circle the two that you are going to use - what you WNT: the side you are solving for (the variable) - what you HVE: a side with a known value 4. Set up your trig ratio (SOH-CH-TO) and solve Eample 1: Use Trigonometric Ratios to Find Side Lengths Find the value of. Round to the nearest tenth Find the lengths of both missing sides. Round to the nearest tenth. 72º 12 TO FIND THE MISSING SIDE OF RIGHT TRINGLE GIVEN 2 SIDES Use the Pythagorean Theorem E: GIVEN 1 SIDE ND 1 NGLE If it s as SPECIL right : 1: : : 2 If it s NOT a special one Use a Trigonometric Ratio E: 63º 11 SOH CH TO! 9

10 Eample 3: Find the value of. If possible leave your answers as a simplified radical or round to the nearest tenth º B. 4 C. D Find. 4. If b = 14 in the right triangle below, then what is the value of c?. 6 B. C. 12 D. 3 6 g 5. Find the value of f and g. 6. Timmy rides up his 12-foot high slanted driveway. g Estimate the length of the driveway. 22 c a b f j 28 h More review Find the sin B and cos B if tan B G 10 I a) Find tan H b) Find cos H C B H 10

11 4.5 More Word Problems + Multiple Choice ngle of ELEVTION: When you look up at an object, the angle that your line of sight makes with a line drawn horizontally. Word Problems Can you say GLORIOUS? ngle of DEPRESSION: When you look down at an object, the angle that your line of sight makes with a line drawn horizontally. Eample 1: Solve the word problems. Round to the nearest tenth. 1. The angle of elevation from the base to the top of a slide is 2. To an observer on a cliff 360 m above sea level, the about 13. The height of the slide is 13.4m. Estimate the angle of depression of a ship is 28. What is the length of the slide. horizontal distance between the ship and the base of the cliff? 3. sonar operator on a cruiser detects a submarine at a distance of 500 m and an angle of depression of 37. How deep is the submarine? foot ladder makes a 76 angle of elevation with the ground. How far up the wall is the top of the ladder? sin 76 cos tan Larry the Ladybug is standing 14 ft from a tall tree. The angle of elevation from the ground to Larry is 43. Find the height of the tree. 11

12 Step Up. 6. Jake s eye level is feet above the ground From Jake s eye level, the angle of elevation is to the top of a building is 29 If the distance from Jake s eye level to the top of the building is 5 feet, how tall is the building? 7. Sally went to the zoo and stood 1 feet away from a giraffe The angle of elevation from Sally s eye level to the top of the giraffe s head is. If the giraffe is 12 feet tall, then find the distance from Sally s eye level to the ground 8. crazy person was chasing Zack down the street The angle of elevation from Zack s eye level to the top of the crazy person s head is. If the crazy person is 6 foot tall and the distance from Zack s eye level to the ground is 4, then how far away is Zack from the crazy person? In groups of Find the perimeter of the triangle below. Round your answer to the tenths place. 2. Find the area of the triangle below. Round your answer to the tenths place

13 How To Handle ll Types of Right Triangle Problems Finding SIDE LENGTHS in a Right Triangle Is it special??? - - or - - YES! NO! : : : : Given: 2 sides lengths Given: 1 angle and 1 side length O H 45 a c O H b RIGHT TRINGLES in POLYGONS O Rectangle w/ a diagonal Square w/ a diagonal H Equilateral w/ altitude H 30 O 30 Classifying a Triangle as RIGHT, CUTE, or OBTUSE Right Triangle Obtuse Triangle cute Triangle HINTS! * Make sure c is the largest number O H legs = hyp legs < hyp legs > hyp a 2 + b 2 = c 2 a 2 + b 2 < c 2 a 2 + b 2 > c 2 * If, then use decimals to check size only 13

14 Eamples 1. Find the value of and y. 2. Solve for. 3. If, then find. H O K 4. Find sin V. 5. Find the height of an equilateral triangle with a side length of 12 U 15 V 6. Solve for W B. C. 6 D. 7. If the diagonal of a square is 10, then find the area of the square. 8. Find the measure of and eplain. R. 100 B. 25 C. 50 D. 200 U. 15 B. 17 C. 120 D. 35 G 14

15 4.6 Inverse Trigonometric Ratios Opening Question: re the questions below asking for the same thing? 30 B 1. Find sin. 2. Find m. 12 C 32 What s an Inverse Operation? Inverse Operations are two operations that undo each other. O H For eample, addition 14and 3 subtraction are inverse operations you use in order to get a variable by itself. 28 E: + 3 = 5 45 To get rid of trig ratios to find an angle use the inverse operations of sine, cosine, and tangent So in general: To solve for ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Eample 1: Find the measure of each given angle. Round to the nearest tenths place. 1. sin tan R Eample 2: Find the measure of B. Round your answer to the nearest tenth. 1. C C B B C 40 B Find the missing sides and angles. Round answers to the nearest tenth

16 Use trigonometric ratios when trying to find a missing side given an angle and a side. E: Trig. Ratios VS. Inverse Trig. Ratios tan 37º = E: 6 6? 37º 3 Use INVERSE trigonometric ratios when trying to find a missing angle given two sides. 8? tan = (tan -1 )tan = (tan -1 ) = tan -1 SOLVE RIGHT TRINGLE To solve a right triangle, find the measures of all and all. When given 1 side & 1 angle (besides the right angle) 1. Pick a reference angle, and use a TRIG RTIO to solve for a missing side. (sin, cos, or tan) 2. Use a TRIG RTIO or PYTHGOREN TH to find the last side. (sin, cos, or tan) OR (a 2 +b 2 =c 2 ) 3. Find the last angle Use TRINGLE SUM TH to find the measure of the missing angle. ( ) When given 2 sides?? 1. Use an INVERSE TRIG RTIO to find the measure of a missing angle. (sin -1, cos -1, or tan -1 ) 2. Use TRINGLE SUM TH to find the measure of the last angle. ( 3. Find the last side Use PYTHGOREN TH to find the missing side (a 2 +b 2 =c 2 )??? (besides the right angle) **Hint!** Start with what you have 2 of. TRIG ratio for what you have 1 of. Eample 1: Solve the right triangle. If possible, leave your answer as a simplified radical or round to the nearest tenth if necessary

17 EXTR PRCTICE If possible, leave your answer as a simplified radical or round to the nearest tenth. 1. Solve a right triangle that has a 30º angle and a 14 inch hypotenuse. (DRW IT!!!) 2. Solve a right triangle with legs that measure 16 and 24 ft. (DRW IT!!!) 17

18 4.7 Spiral Review Parallel lines Corresponding s Consecutive Interior s lternate Interior s lternate Eterior s REFRESH your memory! When lines are Solve for. Eplain. 2. Solve for. Eplain Solve for. Eplain. 4. Solve for. Eplain y Similarity REFRESH your memory! SSS ~ SS ~ ~ S L L S L M L S M S 1. Use ngle-ngle Similarity Postulate to determine which pair of triangles is not similar. a. b. c. d. b c a 2. Use Side-ngle-Side Similarity to determine which pair must be similar. a. b. c. d. 28 M N U D b 8 M 5 c a M N

19 8 Finding Missing 1 st : Write Similarity Statement Sides of Similar 2 nd : Find Ratio of Similarity Triangles 3 rd : Set up proportion & Solve If N = 9, HO= 26, and T = 16, find YO. Do not leave your answer in a decimal. N IF OU = 56, UG = 14, YU = 4, and YG = 18, find DO. Do not leave your answer in a decimal. O T O D Y U H Y G 3. If JM = 12, MS = 15, and M = 18, find ME. Do not leave your answer in a decimal. J E M 4. If VR = 18, VI = 6, and IT = 8, find IC. Do not leave your answer in a decimal. V I C S R T J M E O S Triangle Sum 1. What is the measure of the largest angle of the triangle? ( + 36) ll the interior angles of a triangle add up to Y 2. Find the measure of the smallest angle. G U (4 + 9) (3 17) 19

20 Classifying Scalene Isosceles Equilateral triangles cute Obtuse Right Equiangular Classify the triangle by its angles and sides Converse If (hypothesis), then (conclusion) Countereample True for 1 st part False for 2 nd part 1. If a triangle has two congruent sides, then it is a right triangle. a. Write the converse of the statement b. Determine whether each eample below serves as a countereample. Eplain why for each. E#1: 14, 14, 10 E#2: 6, 6, 6 2. If two angles are acute, then they are congruent. a. Write the converse of the statement b. Determine whether each eample below serves as a countereample. Eplain why for each. E#1: E#2: 20