84 Geometric Mean (PAAP and HLLP) Recall from chapter 7 when we introduced the Geometric Mean of two numbers. Ex 1: Find the geometric mean of 8 and 96.ÿ,. , ...... :£dÿJ In a right triangle, an altitude darn form the vertex of the right angle to the hypotenuse forms two additional ÿJ%c¢ÿS . These triangles from a special relationship. When the altitude is drawn to the hypotenuse of a right triangle, the two smaller triangles are similar to the original. C B x D Y When you drop an altitude like this, you create a geometric mean relationship. By looking at these triangles separately, we can see how each piece is related: C D D B Short Leg_ a x h Hypotenuse _ b Long Leg b Short Leg Hypotenuse cÿa *Notice how certain Long Leg h relationships involve the Geometric Mean! Referring back to our figure: C b B x D Y There are 2 geometric mean relationships you need to know. First is PAAP., Part- Altitude-Altitude - Part Use PAAP when you have information about the altitude. fÿTÿ altitude altitude Second is HLLP, Hypotenuse - Leg-Leg - Part Use HLLP when you have information about the legs. leg **Therefore the ÿ & the altitude are the ONLY geometric means on a right triangle. ALTITUDE GEOMETRIC MEAN THEOREM "PAAP" 4 5 /\ EX: Solve for x, LEG GEOMETRIC MEAN THEOREM "HLLP" × 1 8 IO +jfJÿ EX: Find the length of CD. < 20 ft 15 ft B I I s 2 Cx " <;ÿ 8-2 The Pythagorean Theorem and Its Converse (Applications) The Pythagorean Theorem can be used for many real world application problems. To do an application problem follow the steps listed. 1. Read the problem carefully. 2. Draw and label a diagram to represent the scenario. 3. Write an equation. 4. Solve the equation. 5. Decide if the answer is reasonable. 6. Give the answer units. Example 1: What is the length of a diagonal of a rectangle whose sides measure twelve inches and five inches? (zÿ : ,ÿ,ÿ 25 ÿ(ÿ ÿxa Example 2: A 40-ft ladder is standing 10 feet from a flat roof building and just reaches the top of the building. How tall is the building? X"a' 4-(oÿ "-- Uÿoÿ> b-.foÿ,"ÿ Xÿ --- ('gÿo .-_ J-igg Example 3: A 42 inch TV is for sale. The measurement given is the length of the diagonal across the screen. The aspect ratio of a TV screen is the ratio of the width to the height of the screen. The TV you are thinking of buying has an aspect ratio of 4 : 3. Find the height and width of the 42 inch TV. J ....3z,, %*" ÿ ((oÿÿ _ zs.z;o. = "3 : S" 13 8-2 The Pythagorean Theorem and Its Converse In a right triangle, the following relationship is true. c a a2+ b2= c2 or c2 = a2 -[- b2 *this is true because of the b symmetric property of equality Remember: a and b are legs, c is the hypotenuse and is located across from the Always identify which side is c in your triangle first Example: Identify the unknown side and find its value. Simÿ all radicals (the number under the square root sign must be as small as possible). Use factoring trees. Simplify: 4Yÿ 4Yÿ I\ i\ Z$ls ÿ Z Pythagorean Theorem can be used to find the altitude of an isosceles triangle, allowing us to calculate the area. Formula for area of a triangle is: A= !bh 2 Where b is the lenqth of the base and h is the altitude of the triangle_. When we drop an altitude in an isosceles triangle, the base is bisected into two congruent lengths. Ex 1: 26 26 20 Find the area of the triangle. Pythagorean Triples are a set of three integers which make a2+ b2 = c2 true. Common Triples: 3, 4, 5 3x, 4x, 5x 5,12,13 5x, 12x, 13x 8,15,17 8x, 15x, 17x 7, 24, 25 7x, 24x, 25x If you recognize these triples, you can save yourself a lot of work by skipping the Pythagorean Theorem calculation. Pythagorean Theorem can also be used to show that a triangle is a right triangle. If the relationship is true, then we have a right triangle. Ex 2: Is the following a right triangle? a) 22, 14, 26 2 227 #lqÿ ÿ 2ÿÿ &gO; G?ÿ b) 9, 15, 30-g G I ÿ-zs = q. aoq We can also use Pythagorean Theorem to determine if a triangle is acute or obtuse. If c2 < a2 + b2 then triangle is acute If c2 > a2 + b2 then triangle is obtuse If c2 = a2 + /92 then triangle is ÿt Ex 3: Classify the triangle as acute, obtuse or right. 5, 6, 7 ql zs ÷a(o Ex 4: What type of triangle is made using 4.3ft., 5.2ft., & 7.2ft. 2,2ÿ- t4,ÿÿ +5.2ÿ ÿ ÿ 8-3 Special Tight Triangles In a 45o - 45o - 90o triangle, the legs ÿ are congruent and the length of the hypotenuse h is ÿrÿ times the length of a leg. Symbols: in a 45o - 45o - 90o triangle, ÿ = ÿ and h = ÿr5 . Ratios: the ratio of the sides (leg : leg : hypotenuse) are 1 : 1 : ÿ/-2. X Y E!: Solve for the missing measures. 8 19 In a 30° - 60o - 90o triangle, the length of the hypotenuse h is 2 times the length of the shortest leg s, and the length of the longest leg ÿ is ÿ/3 times the length of the shortest leg. Symbols: in a 30o - 60o - 90o triangle, h = 2s and ÿ = ÿ/3. *Remember, in a triangle the shortest side is always opposite the smallest angle. Which means in a 30o - 60° - 90° triangle, the shortest side s is always opposite of the 30o angle, and the longest side ÿ is always opposite the 60o angle. Ratios: the ratio of the sides (short leg : long leg : hypotenuse) is 1 : ÿ/3 : 2 X E_XX: Solve for the missing measures \lo LL 5 UL Example: The diagonal of a square is 15cm. Find the length of each side. Y ] 8-4 Trigonometry (Inverses) If you know the ratio of the sides (using sine, cosine, or tangent) of a right triangle, then you can use their IÿvÿPÿ to find the measurement of the angle referenced. The followinq are the Inverse Trigonometric Ratios: Ratio Words Symbols If _/.4 is an acute angle and the sin A is Inverse of Sine (sin_l) x, then the inverse sine of x is the If sin A = x, then 5:[ÿ "ÿ (',x'/= Iq, measure of _/`4. If _/.4 is an acute angle and the cos A is Inverse of Cosine (cos_l) x, then the inverse cosine of x is the If cos A = x, then ÿS-ÿ(ÿÿ- ÿ measure of Z`4. Inverse of If _/`4 is an acute angle and the tan A is Tangent x, then the inverse tangent of x is the If tan A = x, then qo.ÿ'r(ÿ:) =/4, (tan-1) measure of _/,4. Examples: Find the measurement of each angle. 4ÿ OÿAÿ ÿ)(:%eÿE C 12 11 5 C. d. ! 8ÿ 4_ISÿ. = yz 1( 6 jiX : X A 1o 31 8-4 Trigonometry Trigonometry, or the study of triangle measurement, involves many types of trigonometric ratios. A trigonometric ratio is a ratio of the lengths of two ÿloÿS of a PÿtrÿHT triangle. The three most common ratios are as follows: Words Symbols If ÿBC is a right triangle with sin A opp 7ÿ6 a. acute ZA , then the sine of ZA hyp tÿ rÿ 6 sin A (written sin A) is the ratio of the sin B length of the leg opposite of ZA (opp) to the length of the sin B opp Pr6 b A hyp 6 6 hypotenuse (hyp). If AABC is a right triangle with cosA-adj- tÿ 6 ÿ b acute ZA , then the cosine of hyp O, G cos A ZA (written cos A) is the ratio of cos B the length of the leg adjacent to b ZA (adj) to the length of the cos B adj 8C, hyp ÿ6 6 hypotenuse (hyp). If A4BC is a right triangle with tanA-°PP- geo. Oc acute ZA , then the tangent of a4 ÿ' b C a B tan A ZA (written tan A) is the ratio of tan B the length of the leg opposite of ZA (opp) to the length of the leg tanB-°PP-=-==Ae" ÿ Iÿ adj ÿ adjacent to ZA (adj). EASY WAY TO REMEMBER YOUR TRIG FUNCTIONS SOH - CAH - TOA SOH - Sine is opposite side over hypotenuse CAH - Cosine is adjacent side over hypotenuse TOA - Tangent is opposite side over adjacent side 25 TBfX Ex 1: Express each ratio as a fraction and a decimal rounded to the nearest hundreth. a. sin P b. cos P Q IS 2 15 c. tan P d. sin Q 17 R 8 8 P e. cosQ f. tanQ I1 lg Our calculators can find this ratio for any angle. **Be sure to be in degree mode!!** Ex 2:Find the following: cos 68o= O. ÿq tab Z60 = B.qCÿ { o.:}?,ÿ{ÿo-ÿ.., x (_ o .'ÿ ÿ7ÿ7...) Ex 3: Find the values of x and y. Ex 4: Find the values of x and y. × × 15 Y ....- s,^r,.ÿ.ÿ.} =- Tÿ 26 uj,ÿ I'Z.5 ÿ Xÿ°lq 8-5 Angles of Elevation and Depression An ÿc€ oF ÿ is the angle formed by a horizontal line and the observer's line of sight to an object above the horizontal line.