SOLUTION: You are given the information that triangle ABC has an altitude of CD. Then you have to write a paragraph proof stating that the Sin A/a = Sin B/b.

Algebra ->  Geometry-proofs -> SOLUTION: You are given the information that triangle ABC has an altitude of CD. Then you have to write a paragraph proof stating that the Sin A/a = Sin B/b.      Log On


   

Question 855017: You are given the information that triangle ABC has an altitude of CD. Then you have to write a paragraph proof stating that the Sin A/a = Sin B/b.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
If angles A and B are acute, altitude CD divides the triangle, into two right triangles:

In those right triangles, the trigonometric ratios involving altitude CD and angles A and B are
sin%28A%29=CD%2Fb<-->CD=b%2Asin%28A%29
sin%28B%29=CD%2Fa<-->CD=a%2Asin%28B%29

If one of the angles A and B is obtuse, sides AC and BC, along with altitude CD, and the line containing AB form two right triangles:
Ray BD is the extension of AB. Obtuse angle ABC is called angle B for short.
Angle B and angle CBD are supplementary and therefore have the same sine.
In right triangles ADC and BDC, the trigonometric ratios involving altitude CD and angles A and B are
sin%28A%29=CD%2Fb<-->CD=b%2Asin%28A%29
sin%28B%29=sin%28CBD%29=CD%2Fa<-->CD=a%2Asin%28B%29

In either case, CD=b%2Asin%28A%29 and CD=a%2Asin%28B%29 ,
so b%2Asin%28A%29=a%2Asin%28B%29
Dividing both sides of the equal sign by ab we get
sin%28a%29%2Fa=sin%28B%29%2Fb