6.4 Inequalities for one triangle

THEOREM THAT COVERS ALL THE CONCEPTS IN 6.4:

-If the angles of a triangle are distinct, then you can draw conclusions based on the side lengths.

CONCLUSIONS SHOWN BELOW


(diagram is not to scale)

GIVEN: m^A=40; m^B=75; m^ C=65









CONCLUSIONS:

~m^A < m^C < m^B

~a < c < b ->because of the theorem If one angle of a triangle is larger than a second, then the side opposite the first angle is longer than the side opposite the second angle.


BASICALLY:

~the smallest angle is opposite the shortest side

~the "middle" angle is opposite the "middle" side

~the largest angle is opposite the longest side







REMEMBER!!!!!!!!!!!!!!!!!!!!

In order for a triangle to exist, the sum of any 2 sides MUST always be GREATER than the 3rd side. (also called the Triangle Inequality Theorem)



Example:

If two sides of a "triangle" are 7 cm long and the third is 16 cm long, then the "triangle" is NOT a triangle.

WHY???

-7+7= 14 < 16

-7+16=23 > 7

The sum of the second pair of side lengths IS greater than the third side, BUT the sum of the first pair of side lengths is NOT greater than the third side, therefore, the "triangle" is NOT a triangle.



ANOTHER EXAMPLE:

If two sides of a "triangle" are 8 cm long and the third side is 14 cm long, then the "triangle" IS a triangle.



WHY???

-8+8=16 > 14

-8+14=22 > 8

The sum of the first pair of side lengths IS greater than the third side, and the sum of the second pair of side lenths IS greater than the third side, therefore, the "triangle" IS a triangle.





COROLLARIES:

-The perpendicular segment from a point to a line is the shortest segment from the point to the line.

-The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.



HELPFUL VIDEO:
-http://www.winpossible.com/lessons/Geometry_Triangle_Inequality.html
-http://www.youtube.com/watch?v=e1prq6UFxJE&noredirect=1 (this is a YouTube video so you gotta watch it at home, but its about Indirect proofs and all of 6.4)