CHAPTER 8 By: Fiona Coupe, Dani Frese, and Ale Dumenigo

8-1 Similarity in right triangles Rt similarity- if the altitude is drawn to the hypotenuse of the triangle then the two small triangles are similar to each other Corollary 1- when the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse Corollary 2- when the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segments of the hypotenuse that is adjacent to that leg – Geometric mean- average in a geometric shape – Altitude- a line from a vertex of a triangle perpendicular to the opposite side

C B N A 1.ACB ~ ANC by AA~ Proportional sides AB = AC AC AN AC is the geometric mean between AB and AN 2.ACB ~ CNB by AA~ Proportional sides AB = BC BC NB BC is the geometric mean between AB and NB 3.ANC ~ CNB Proportional sides AN = CN CN NB CN is the geometric mean between AN and NB EXAMPLE FOR GEOMETIRC MEAN:

H R J E Find HJ, RE, RH and HE RE= 9+16 RE = 25 HJ is the geometric mean between EJ and JR HJ = 9 = HJ HJ 16 HJ 2 = 144 HJ = 12 RH is the geometric mean between RE and JR RH = 25 = RH RH 16 RH 2 = 400 RH = 20 HE is the geometric mean between EJ and ER HE = 9 = HE HE 25 HE 2 = 225 HE = 15 EXAMPLE FOR GEOMETRIC MEAN #2

Y X A Z If XZ = 36, AX = 12, and ZY = 49 find ZA, YZ, YX GEOMETRIC MEAN EXAMPLE #3

8.2 Pythagorean Theorem If sides a and b are the legs of a right triangle and c is the hypotenuse then… a 2 + b 2 = c 2

Examples: 1.A=2 B=3 and C=x2. A=x B=x C=4 2²+3²= x²x² + x²= =x²2x²=16 √13=x2x²/2= 16/2 x²=8 x= √8 = 2√2 3. Find the diagonal of a rectangle with length 8 and width 4 8²+4²=c² 64+16= c² 80=c² √80=4√5=c 4 8

8.3 Converse of Pythagorean Theorem c²= a²+b² then the triangle is right c²< a²+b² then the triangle is acute c²> a²+b² then the triangle is obtuse

Examples: Sides: 6,7,8 Start by comparing the longest side to the shorter ones 8²= 64 6²+7²= =85 64 < 85 The triangle is acute

Common right triangles A=3 B=4 C=5 A=5 B=12 C=13 A=8 B=15 C=17 A=7 B=24 C=25 These common right triangles also apply for their multiples

8-4 Special Right Triangles Theorem in a triangle the hypotenuse is times as long as a leg Theorem 8-7- in a triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg

A A A A 2A A Find the missing sides for the two triangles X X X = 7 12 A B C AC = 6 CB = 6 EXAMPLES:

Solve for the missing sides A B C X Y Z

Trigonometry Tangent Ratio The word trigonometry comes from Greek words that mean “Triangle measurement.” Tangent Ratio: Tangent of <A = leg opposite <A Opposite Leg Adjacent LegA leg adjacent <A In abbreviated form: tan A = opposite adjacent Example: Find tan X and tan Y. Y ZX Tan X= leg opposite <X = 12 leg adjacent to <X = 5 Tan Y= leg opposite <Y = 5 leg adjacent to <Y = 12

Tangent Example Example: Find the value of Y to the nearest tenth Y 56° 32 Tan 56° = y 32 Solution: y= 32(tan 56°) Y= 32(1.4826) Y= or 47.4 Workout Problem: Find x in this right triangle: x 38° 46

The Sine and Cosine Ratio The ratios that relate the legs to the hypotenuse are the sine and cosine ratios. Sine of <A= leg opposite <A hypotenuse Cosine of <A= leg adjacent to <A hypotenuse Adjacent Leg Opposite Leg A hypotenuse Find value of x and y to the nearest integer ° x y Sin 67 ° = x/120 X= 120 ∙ sin 67 ° X= 120(0.9205) X= or 110 Cos 67 °= y/120 Y= 120 ∙ cos 67 ° Y= 120(0.3907) Y= or 47 State 2 different equations you could use to find the value of x. 49 ° x 41 ° 100

SOHCAHTOA SOH- sine (the angle measurement)= opposite leg/ hypotenuse CAH- Cosine (the angle measurement) = Adjacent leg/ hypotenuse TOA- Tangent (the angle measurement) = Opposite leg/Adjacent

Applications of Right Triangle Trigonometry The angle between the top horizontal and the line of sight is called an angle of depression. An angle of elevation is the angle between the bottom horizontal and the line of sight. Angle of elevation: 2° Angle of depression 2° Horizontal x If the top of the lighthouse is 25 m above sea level, the distance x between the boat and the base of the lighthouse can be found in 2 ways. Tan 2° = 25/x X= 25/ tan 2° X= 25/ X= Tan 88°= x/25 X= 25(tan 88°) X= 25( ) X= A good answer would be that the boat is roughly 700 m. from the lighthouse

Examples A kite is flying at an angle of elevation about 40°. All 80 m of the string have been let out. Ignoring the sag in the string, find the height of the kite to the nearest 10 m. Two buildings on opposite sides of a street are 40 m. apart. From the top of the taller building, which is 185 m high, the angle of depression to the top of the shorter building is 13°. Find the height of the shorter building. 40° 80 x Sin 40° = x/80 X= ° 40 x 185 Tan 13° = x/40 X= =