SOLUTION: In the diagram below, O is the centre of the circle and OS is perpendicular to the chord RT Prove the theorem that states that RS=ST

Algebra ->  Geometry-proofs -> SOLUTION: In the diagram below, O is the centre of the circle and OS is perpendicular to the chord RT Prove the theorem that states that RS=ST      Log On


   

Question 1123083: In the diagram below, O is the centre of the circle and OS is perpendicular to the chord RT
Prove the theorem that states that RS=ST
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
In the diagram below,
-------
How far below?

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
The triangle ORT is an isosceles triangle, having the lateral sides OR and RT congruent (since they are the radii of the circle). 


The segment OS is the height in the triangle ORT drawn to its base.


In an isosceles triangle the altitude drawn to the base is the median in the same time.


Therefore, RS = ST.

Proved.

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See the lesson
    - An altitude a median and an angle bisector in the isosceles triangle
in this site.