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CHAPTER 8

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

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CHAPTER 8

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  1. CHAPTER 8 By: Fiona Coupe, DaniFrese, and Ale Dumenigo

  2. 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

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

  4. H EXAMPLE FOR GEOMETRIC MEAN #2 RE= 9+16 RE = 25 HJ is the geometric mean between EJ and JR HJ = 9 = HJ HJ 16 HJ2 = 144 HJ = 12 RH is the geometric mean between RE and JR RH = 25 = RH RH 16 RH2 = 400 RH = 20 HE is the geometric mean between EJ and ER HE = 9 = HE HE 25 HE2 = 225 HE = 15 12 R E J 9 16 Find HJ, RE, RH and HE

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

  6. 8.2 Pythagorean Theorem If sides a and b are the legs of a right triangle and c is the hypotenuse then… a2 + b2 = c2

  7. Examples: • A=2 B=3 and C=x 2. A=x B=x C=4 2²+3²= x²x² + x²= 16 4+9=x² 2x²=16 √13=x 2x²/2= 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. 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

  9. Examples: • Sides: 6,7,8 • Start by comparing the longest side to the shorter ones • 8²= 64 • 6²+7²= 36+49 =85 • 64 < 85 • The triangle is acute

  10. 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 • Thesecommonright triangles also apply for their multiples

  11. 8-4 Special Right Triangles • Theorem 8-6 - in a 45-45-90 triangle the hypotenuse is times as long as a leg • Theorem 8-7- in a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg

  12. EXAMPLES: 30 45 A 2A A A 45 60 A A Find the missing sides for the two triangles A 60 45 12 7 X 45 30 C X B AC = 6 CB = 6 X = 7

  13. Solve for the missing sides X A 30 45 9 18 60 45 Y Z B C

  14. TrigonometryTangent Ratio • The word trigonometry comes from Greek words that mean “Triangle measurement.” • Tangent Ratio: Tangent of <A = leg opposite <A leg adjacent <A Opposite Leg In abbreviated form: tan A = opposite adjacent A Adjacent Leg Example: Find tan X and tan Y. Tan X= leg opposite <X = 12 Y leg adjacent to <X = 5 Tan Y= leg opposite <Y = 5 12 13 leg adjacent to <Y = 12 Z X 5

  15. Tangent Example Example: Find the value of Y to the nearest tenth Y Tan 56° = y 32 56° 32 Solution: y= 32(tan 56°) Y= 32(1.4826) Y= 47.4432 or 47.4 Workout Problem: Find x in this right triangle: x 38° 46

  16. 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 Cosine of <A= leg adjacent to <A hypotenuse hypotenuse hypotenuse Opposite Leg A Adjacent Leg Find value of x and y to the nearest integer. Sin 67 ° = x/120 X= 120 ∙ sin 67 ° X= 120(0.9205) X= 110.46 or 110 State 2 different equations you could use to find the value of x. 49 ° 120 x 100 x 67 ° Cos 67 °= y/120 Y= 120 ∙ cos 67 ° Y= 120(0.3907) Y= 46.884 or 47 y 41 °

  17. 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

  18. 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 depression 2° Horizontal x Angle of elevation: 2° 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= 25/0.0349 X= 716.3 Tan 88°= x/25 X= 25(tan 88°) X= 25(28.6363) X= 715.9 A good answer would be that the boat is roughly 700 m. from the lighthouse

  19. 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. Sin 40° = x/80 X= 51.4 80 x 40° 40 13° Tan 13° = x/40 X= 9.23 185-9.23= 175.77 x 185 185

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