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Special Right Triangles

  1. 1. Special Right Triangles<br />Geometric Wonder Children<br />
  2. 2. Triangles<br />A triangle is any polygon with 3 sides and 3 angles.<br />Angles must add up to 180º<br />50<br />65<br />65<br />
  3. 3. β<br />90<br />α<br />But something happens…<br />What if one angle was perpendicular, aka, 90º?<br />
  4. 4. Then that means the others have to measure to 90º as well. <br />30+60=90<br />… Wait a minute…<br />30<br />90<br />60<br />
  5. 5. Hypotenuse is always opposite the R. Angle.<br />Some Definitions<br />Hypotenuse<br />Side/Height<br />Side/Base<br />
  6. 6. So special<br />There are different kinds of right triangles:<br />Scalene/30-60-90 <br />Right isosceles/<br />45-45-90<br />Scalene<br />
  7. 7. Pythagoras<br />One really smart dude, Pythagoras, studied really hard.<br />Found this pretty fundamental theorem:<br />Adding the squares of each side-length of a right triangle will equal the square of the hypotenuse.<br />Or: a2+b2=c2<br />
  8. 8. This what that looks like:<br />c2<br />c<br />a2<br />a<br />b<br />b2<br />
  9. 9. There is some consistency with angles and sides<br />Once you know two sides, you can figure out the third<br />32+42=x2<br /> 9+16=x2<br /> 25=x2<br /> 5=x<br />Why is this great?<br />x<br />3<br />4<br />
  10. 10. Special Right Triangles<br />30-60-90<br />Ratios are the same for all lengths<br />45-45-90<br />Ratios are the same for all lengths<br />
  11. 11. 30-60-90<br />Note when the angle is the same…<br />… The lengths of the sides have the same ratios!<br />30<br />30<br />2<br />4<br />√3<br />2√3<br />60<br />1<br />60<br />2<br />
  12. 12. Same is true for 45-45-90!<br />45<br />2√2<br />45<br />1.5√2<br />2<br />1.5<br />45<br />45<br />1.5<br />2<br />Coincidence..? I think not…<br />
  13. 13. For any triangle whose angles are 30-60-90:<br />The shortest side will be half of the length of the hypotenuse and the second longest side will equal to the length of the shortest side times the square root of 3. <br />THIS IS ALWAYS TRUE FOR A 30-60-90 Δs!!<br />Let’s generalize this:<br />
  14. 14. For any triangle with 45-45-90 angles:<br />The length of the hypotenuse will be equal to the length of either side times the square root of 2.<br />THIS IS ALWAYS TRUE FOR 45-45-90 Δs!<br />Similar for 45-45-90:<br />
  15. 15. Right triangles have one fixed 90º angle; the other two angle have to equal 90-x and x, respectively.<br />Ratios of 30-60-90 and 45-45-90 R. triangles are constant.<br />In right triangles, Pythagoras’ theorem is always true: <br />a2+b2=c2<br />What we’ve learned:<br />
  16. 16. Sine<br />Cosine<br />Tangent<br />SohCahToa<br />Pythagorean Triples<br />Next week:<br />"Without geometry life is pointless.” -Anonymous<br />
  17. 17. Powerpoint Auto Shapes<br />Lang, S. & Murrow, G (1983). Geometry: a high school course. New York: Springer-Verlag.<br />References<br />

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