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Right Triangles with SOund on 際際滷 9

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Darren Sowards
Darren Sowards

1) Special right triangles have specific angle measurements (30-60-90 or 45-45-90) that result in consistent side length ratios. 2) The Pythagorean theorem, a2 + b2 = c2, always applies to right triangles and relates the sides. 3) For 30-60-90 triangles, the shortest side is half the hypotenuse, and the second longest side equals the shortest side times the square root of 3. For 45-45-90 triangles, the hypotenuse equals either leg times the square root of 2.

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Special Right TrianglesGeometric Wonder Children
TrianglesA triangle is any polygon with 3 sides and 3 angles.Angles must add up to 180尊506565
硫90留But something happensWhat if one angle was perpendicular, aka, 90尊?
Then that means the others have to measure to 90尊 as well. 30+60=90 Wait a minute309060
Hypotenuse is always opposite the R. Angle.Some DefinitionsHypotenuseSide/HeightSide/Base
So specialThere are different kinds of right triangles:Scalene/30-60-90 Right isosceles/45-45-90Scalene
PythagorasOne really smart dude, Pythagoras, studied really hard.Found this pretty fundamental theorem:Adding the squares of each side-length of a right triangle will equal the square of the hypotenuse.Or: a2+b2=c2
This what that looks like:c2ca2abb2
There is some consistency with angles and sidesOnce you know two sides, you can figure out the third32+42=x2 9+16=x2 25=x2 5=xWhy is this great?x34
Special Right Triangles30-60-90Ratios are the same for all lengths45-45-90Ratios are the same for all lengths
30-60-90Note when the angle is the same The lengths of the sides have the same ratios!303024323601602
Same is true for 45-45-90!4522451.5221.545451.52Coincidence..? I think not
For any triangle whose angles are 30-60-90: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. THIS IS ALWAYS TRUE FOR A 30-60-90 s!!Lets generalize this:
For any triangle with 45-45-90 angles:The length of the hypotenuse will be equal to the length of either side times the square root of 2.THIS IS ALWAYS TRUE FOR 45-45-90 s!Similar for 45-45-90:
Right triangles have one fixed 90尊 angle; the other two angle have to equal 90-x and x, respectively.Ratios of 30-60-90 and 45-45-90 R. triangles are constant.In right triangles, Pythagoras theorem is always true: a2+b2=c2What weve learned:
SineCosineTangentSohCahToaPythagorean TriplesNext week:"Without geometry life is pointless. -Anonymous
Powerpoint Auto ShapesLang, S. & Murrow, G (1983). Geometry: a high school course. New York: Springer-Verlag.References

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Right Triangles with SOund on 際際滷 9

  • 2. TrianglesA triangle is any polygon with 3 sides and 3 angles.Angles must add up to 180尊506565
  • 3. 硫90留But something happensWhat if one angle was perpendicular, aka, 90尊?
  • 4. Then that means the others have to measure to 90尊 as well. 30+60=90 Wait a minute309060
  • 5. Hypotenuse is always opposite the R. Angle.Some DefinitionsHypotenuseSide/HeightSide/Base
  • 6. So specialThere are different kinds of right triangles:Scalene/30-60-90 Right isosceles/45-45-90Scalene
  • 7. PythagorasOne really smart dude, Pythagoras, studied really hard.Found this pretty fundamental theorem:Adding the squares of each side-length of a right triangle will equal the square of the hypotenuse.Or: a2+b2=c2
  • 8. This what that looks like:c2ca2abb2
  • 9. There is some consistency with angles and sidesOnce you know two sides, you can figure out the third32+42=x2 9+16=x2 25=x2 5=xWhy is this great?x34
  • 10. Special Right Triangles30-60-90Ratios are the same for all lengths45-45-90Ratios are the same for all lengths
  • 11. 30-60-90Note when the angle is the same The lengths of the sides have the same ratios!303024323601602
  • 12. Same is true for 45-45-90!4522451.5221.545451.52Coincidence..? I think not
  • 13. For any triangle whose angles are 30-60-90: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. THIS IS ALWAYS TRUE FOR A 30-60-90 s!!Lets generalize this:
  • 14. For any triangle with 45-45-90 angles:The length of the hypotenuse will be equal to the length of either side times the square root of 2.THIS IS ALWAYS TRUE FOR 45-45-90 s!Similar for 45-45-90:
  • 15. Right triangles have one fixed 90尊 angle; the other two angle have to equal 90-x and x, respectively.Ratios of 30-60-90 and 45-45-90 R. triangles are constant.In right triangles, Pythagoras theorem is always true: a2+b2=c2What weve learned:
  • 17. Powerpoint Auto ShapesLang, S. & Murrow, G (1983). Geometry: a high school course. New York: Springer-Verlag.References