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Math Help - Help! 2 questions - triangles and circles

  1. #1
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    Help! 2 questions - triangles and circles

    Please help with these two questions.

    1) In the figure, two circles touch each other at C. AB is a common tangent touching the circles at A and B and angle ABC is 35 degrees. Find the angle CAB.


    2) In the figure, O is the centre of the circle. A and B are two points on the circle so that triangle OAB is an equilateral triangle. OA is produced to C so that OA = AC. Find the angle ABC and prove whether CB is tangent to the circle at B, giving a reason.


    Thankyou so much for any help, I don't know where to start in solving these.
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  2. #2
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    Hello, BG5965!

    2) In the figure, O is the centre of the circle.
    A and B are two points on the circle so that triangle OAB is an equilateral triangle.
    OA is produced to C so that OA = AC.
    Find the angle ABC and prove whether CB is tangent to the circle at B.

    Let r \;=\;AO = BO, radius r.
    . . Then: . CO = 2r

    In \Delta COB,\;\angle COB = 60^o,\;BO = r,\;CO = 2r

    Then: . \sin(\angle C) \:=\:\frac{r}{2r} \:=\:\frac{1}{2} \quad\Rightarrow\quad \angle C \:=\:30^o

    Hence: . \angle OBC = 90^o and OB is tangent to the circle.


    Since \angle OBA = 60^o, then \angle ABC = 30^o

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