SOLUTION: The points A and C lie on a circle with center O and radius 5 sq.root of 2. The point within the circle is such that ABC = 90 degrees. Following the data: AB = 6 BC = 2 Find O

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Question 1081672: The points A and C lie on a circle with center O and radius 5 sq.root of 2. The point within the circle is such that ABC = 90 degrees. Following the data:
AB = 6
BC = 2
Find OB.
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
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The points A and C lie on a circle with center O and radius 5 sq.root of 2. The point within the circle is such that ABC = 90 degrees.
Following the data: AB = 6, BC = 2, Find OB.
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It seems this problem is slightly above the average school geometry level.
So, I will assume that your level corresponds to that.
Therefore, I will give you only general instructions/directions, leaving calculations and details to you. 

0.  Make a sketch to follow my arguments.

    You have the circle of the radius 5%2Asqrt%282%29 with the center at the point O.

    You have the right-angled triangle ABC leaning on the chord AC.
    The legs AB and BC are of the lengths 6 and 2 units.


1.  Hence, the length of the chord AC (which is the hypotenuse) is

    |AC| = sqrt%286%5E2%2B2%5E2%29 = sqrt%2840%29.

2.  The distance from the center O to the chord AC is sqrt%28R%5E2+-+%28abs%28AC%29%2F2%29%5E2%29 = sqrt%28%285%2Asqrt%282%29%29%5E2+-+%28sqrt%2840%29%2F2%29%5E2%29 = sqrt%2850-10%29 = sqrt%2840%29.

    It is the length of the perpendicular OD drown from the center O to the mid-point D of the chord AC.


3.  Draw the altitude BE in the right angled triangle ABC from the right angle vertex to the hypotenuse AC.
    (E is the base of this altitude).

    You can find the length of BE from the "AREA" equation of the triangle ABC:

    %281%2F2%29%2Aabs%28AC%29%2Aabs%28BE%29 = %281%2F2%29%2A2%2A6,   ====>  |BE| = 12%2Fsqrt%2840%29.


4.  Now you can calculate one component of the segment OB.

    (I call this component "vertical" component of the segment OB, since it is vertical component in my sketch . . . )

    This vertical component is  |OD| - |BE| = sqrt%2840%29+-+12%2Fsqrt%2840%29.


5. Next step is to find the horizontal component of the segment OB.

   For it, you need to determine in which segments the altitude DE divide the hypotenuse AC.

    It is very standard task (sub-task), and every advanced student must be able to do it.
        (Use similarity of right-angled triangles).

    When you complete this sub-task, you will get the horizontal component of the segment OB by subtracting half of the chord AC length.


6.  Having the horizontal and the vertical components of OB, you will be in position to find its length.