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Triangles

Triangles. Classifications of Triangles Sum of Angles in triangles Pythagorean Theorem Trig Ratios Area of Triangles. Triangle Review. Acute Triangle Obtuse Triangle Right Triangle Equilateral Triangle Isosceles Triangle Scalene Triangle Sum of the angles in a triangle is 180.

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Triangles

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  1. Triangles Classifications of Triangles Sum of Angles in triangles Pythagorean Theorem Trig Ratios Area of Triangles

  2. Triangle Review • Acute Triangle • Obtuse Triangle • Right Triangle • Equilateral Triangle • Isosceles Triangle • Scalene Triangle • Sum of the angles in a triangle is 180

  3. Solve for x and classify the triangles by its sides and angles 5x 3x 10 10 3x y+7 2x 55 10 3x=45 y+7=45 x=15 y=38 Isosceles Right Triangle 5x+55=180 5x=125 X=25 75,50,55 Scalene, Acute 5x=60 x=12 Equilateral, Acute

  4. c b a The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. a2+ b2=c2

  5. 8m 8m h 5 5 10m Example 1: Find the Area Because the triangle is isosceles, the base is bisected. Use pyth. Thm to find “h” a2+b2=c2 52+h2=82 25+h2=64 h2=39 h=6.24 Area of Triangle A=1/2 bh A=1/2(10)(6.24) A=31.2m2 =6.24

  6. 2nd base 65ft 65ft 3rd base Pitcher’s Mound 91.9ft 1st base 50ft 65ft 65ft Home Plate Example 2: In slow pitch softball, the distance between consecutive bases is 65 ft. The pitcher’s plate is located on the line between second based and home plate 50 ft from home plate. How far is the pitcher’s plate from second base? Justify your answer You can usethe Pyth. Thm: a2+b2=c2 652+652 = x2 4225+4225=x2 8450 = x 2 91.9ft from home plate to 2nd base = x Total distance - PM to HP = 2nd to PM 91.9 - 50 = 41.9ft

  7. Pythagorean Triple: 3 positive integers a,b,c, that satisfy a2+b2=c2 Example: 3,4,5 represent a Pythagorean Triple 32+42=52 9+16 = 25 25=25

  8. Common Pyth Tripples

  9. B c a A C b Trig Ratios: Sin A = Opposite Side Hypotenuse Cos A = Adjacent Side Hypotenuse Tan A = Opposite Side Adjacent Side Hypotenuse Opposite Adjacent

  10. B Hypotenuse 5 3 A C 4 Adjacent Example 1: Find the ratio of the sin A, cos A and Tan A Sin A = Opp Hyp. = 3 5 = 4 5 Cos A = Adj Hyp Opposite Tan A = Opp Adj = 3 4

  11. B 5 3 A C 4 Example 2: Find the ratio of the sin B, cos B and Tan B Sin B = Opp Hyp. = 4 5 Hypotenuse = 3 5 Cos B = Adj Hyp Adjacent Tan B = Opp Adj = 4 3 Opposite

  12. Sin, Cos and Tan on your Calculator Use your calculator: Cos 13º = _______ .9744 .4540 Sin 27º = _______ 2.2460 Tan 66º = _______

  13. x 48º 100 ft Example 4: Find the height of the silo. Tan 48 = x 100 Solve by cross mult. You can use Tan Ratio: Tan A = opp adj X = 100 ● tan 48 X = 111 ft.

  14. x 38º 154 ft Example 5: You are measuring the height of a tower. You stand 154 ft. from the base of the tower. You measure the angle of elevation from a point on the ground to the top of the tower to be 38º. Estimate the height of the tower. Tan 38 = x 154 Tan A = opp adj X = 154 ● tan 38 X = 120 ft.

  15. 14 9 47 x x 15 x 22 35 19 52 Example 6: Other Variations: Solve for x Sin 22 = x/14 14Sin22 = x Cos 47 = 9/x xCos 47 = 9 X = 9/cos47 Sin 35 = 15/x xSin35 = 15 X = 15/Sin35 Cos 52 = x/19 19Cos 52 = x x

  16. B 10 8 C A x Example 7:Solve the Right Triangle Sides: AB = 8 BC = 10 Missing AC: To find AC use pyth thm 82+102=x2 64+100 = x2 164 = x2 12.8 = x, AC = 12.8

  17. B adj 10 8 opp C A x Example 7:Solve the Right Triangle Angles: <B = 90 Missing <A and <C: To find find the missing angles, we will use INVERSE trig functions. To get A by itself, we must do the opposite of Tan. This is called INVERSE TAN, it is Tan-1on your calculator Tan A = 10 8 Tan A = opp adj Tan A = 1.25 Tan-1Tan A = Tan-1 1.25 A = 51.34º

  18. B adj opp 10 8 C A x Example 7:Solve the Right Triangle Angles: <B = 90 <A = 51.34 Missing <C: To find find the missing angles, we will use INVERSE trig functions. To get A by itself, we must do the opposite of Tan. This is called INVERSE TAN, it is Tan-1on your calculator Tan C = 8 10 Tan C = opp adj Tan C = .8 Tan-1Tan C = Tan-1 .8 C = 38.66

  19. B 10 8 C A x Example 7:Solve the Right Triangle Sides: AB = 8 BC = 10 AC = 12.8 Angles: m<B = 90 m<A = 51.34 m<C = 38.66

  20. 15 12 50 14 14 18 10 Example 8: Find the area of each triangle

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