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CHAPTER 11 The Right Triangle

CHAPTER 11 The Right Triangle. By: Jimmy Sun and Ben Gillingham. 11.1 Proportions of right triangles. Arithmetic mean- x+y/2 Geometric mean- a/b= b/c Geometric mean uses proportions in finding various segments of a triangle. Geometric Mean Theorems.

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CHAPTER 11 The Right Triangle

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  1. CHAPTER 11 The Right Triangle By: Jimmy Sun and Ben Gillingham

  2. 11.1 Proportions of right triangles Arithmetic mean- x+y/2 Geometric mean- a/b= b/c Geometric mean uses proportions in finding various segments of a triangle

  3. Geometric Mean Theorems Hypotenuse- Side of the triangle opposite of the right angle The altitude to the hypotenuse of a right triangle forms two triangles similar to it and each other. The altitude to the hypotenuse of a right triangle is the geometric mean between the segments into which it divides the hypotenuse. Each leg of a right triangle is the geometric mean between the hypotenuse and its projection on the hypotenuse -Projection means adjacent leg.

  4. Pythagorean Theorem In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. A squared + B squared = C squared C A B

  5. Special Right Triangles There are two special right triangles in this chapter Isosceles Right Triangle- Both legs equal and they form two 45 degree angles 30-60 Right Triangle- The angles in the triangle form angles of 30, 60, and 90 degrees

  6. Other corollaries Each diagonal of a square is the square root of 2 multiplied by the length of its side. The altitude of an equilateral triangle having side s is s*square root of 3 , and its area is s^2 * square root of 3 2 4

  7. REMEMBER The formula for the area of an equilateral triangle is very useful It can be used to find the area of a regular hexagon if one side is known, since a hexagon is composed of 6 equilateral triangles

  8. 11.2 the Pythagorean Theorem 11.3 (30,60,90) (45,45,90) triangles

  9. PYTHAGOREAN TRIPLES -3,4,5 -6,8,10 -9,12,15 -12,16,20 -15,20,25 -5,12,13, -10, 24, 26 -7,24,25 -8,15,17 -20,21,29 -9,40,41

  10. Slope The slope of a line is equivalent to rise over run Rise= change in y Run= Change in x DO NOT FORGET: -Parallel lines have EQUAL slopes and perpendicular lines have opposite of reciprocal slopes. -arctan(slope)= angle of inclination

  11. TRIGONOMETRY: SOH CAH TOA Theta is the angle Sine:sin(Theta)= Opposite/Hypotenuse Cosine: cos(Theta)= adjacent/hypotenuse Tangent: tan(Theta)= opposite/adjacent note: finding angles Theta=sin^-1(O/H)=cos^-1(A/H)=tan^-1(O/A)

  12. 11.7 1/2 Law of Sine side a/sin(A) = side b/sin(B) = side c/sin(C) C b a A B c

  13. 11.7 2/2 Law of Cosine Law of Cosines can be used to find specific angles and sides in any type of triangle A^2 = B^2 + C^2 - 2BCcos(a) B^2 = A^2 + C^2 - 2ACcos(b) C^2 = A^2 + B^2 - 2ABcos(c) B c A a b C

  14. Practice Problem 1 A Given that angle ACB is a right angle and that CX divides triangle ABC into two right triangles, find CX and XB. AX= 21BC= 10 21 X B C 10

  15. SOLUTIONS PP1) XB=4 CX =2 rad 21

  16. Summary Remember- Geometric mean can be used to find side lengths of two right triangles. Given two sides of a right triangle, the Pythagorean Theorem can be used to find the third side. Know the relationships between the sides in the isosceles right triangle and the 30-60-90 triangle. Remember SOHCAHTOA and its uses. Remember that parallel lines have equal slopes and perpendicular lines have opposite of the reciprocal slopes. Also, remember formulas for law of sines and law of cosines.

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