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 3x=45y+7=45 x=15y=38 Isosceles Right Triangle 5x=60 x=12 Equilateral, Acute 5x+55=180 5x=125 X=25 75,50,55 Scalene, Acute 3xy+7 5x 552x 3x 10

4. 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. a 2 + b 2 =c 2 a b c

5. Example 1: Find the Area Because the triangle is isosceles, the base is bisected. 8m 10m h Use pyth. Thm to find “ h ” a 2 +b 2 =c 2 5 2 +h 2 =8 2 25+h 2 =64 h 2 =39 h=6.24 Area of Triangle A=1/2 bh A=1/2(10)(6.24) A=31.2m 2 =6.24 55

6. Pitcher ’ s Mound 65ft Home Plate 1st base 2nd base 3rd base 50ft 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 use the Pyth. Thm: a 2 +b 2 =c 2 65 2 +65 2 = x 2 4225+4225=x 2 8450 = x 2 91.9ft from home plate to 2nd base = x 91.9ft Total distance - PM to HP = 2nd to PM 91.9 - 50 = 41.9ft

7. Pythagorean Triple: 3 positive integers a,b,c, that satisfy a 2 +b 2 =c 2 Example: 3,4,5 represent a Pythagorean Triple 3 2 +4 2 =5 2 9+16 = 25 25=25

8. Common Pyth Tripples

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

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

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

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

13. Example 4: Find the height of the silo. 48º 100 ft x You can use Tan Ratio: Tan A = opp adj Tan 48 = x 100 S o l v e b y c r o s s m u l t. X = 100 ● tan 48 X = 111 ft.

14. 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. 38º 154 ft x Tan A = opp adj Tan 38 = x 154 X = 154 ● tan 38 X = 120 ft.

15. Example 6: Other Variations: Solve for x 14 22 x 479x 35 15 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 52 19 x Cos 52 = x/19 19Cos 52 = x

16. Example 7: Solve the Right Triangle A B C 8 10 x Sides: AB = 8 BC = 10 Missing AC: To find AC use pyth thm 8 2 +10 2 =x 2 64+100 = x 2 164 = x 2 12.8 = x, AC = 12.8

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

18. Example 7: Solve the Right Triangle A B C 8 10 x Angles: <B = 90 <A = 51.34 Missing <C: To find find the missing angles, we will use INVERSE trig functions. Tan C = opp adj Tan C = 8 10 Tan C =.8 To get A by itself, we must do the opposite of Tan. This is called INVERSE TAN, it is Tan -1 on your calculator Tan -1 Tan C = Tan -1.8 C = 38.66 opp adj

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

20. Example 8: Find the area of each triangle 14 10 50 1812 15