Geometry of a Circle (Arcs and Angles)

Date: 10/13/97 at 15:26:33
From: Silmarie Nieves
Subject: Geometry of a circle (arcs and angles)

DE is a diameter of circle O, and is perpendicular to chord AB at 
point C. 

Let AB = 24, CD = 8, and OC = X

   A. Represent the length of OB in terms of X.
   B. Using OB, OC, and CB, write an equation that can be used 
      to find X.
   C. Find OC.
   D. Find OB.

I don't understand this problem. Please help.

Date: 10/16/97 at 18:05:56
From: Doctor Chita
Subject: Re: Geometry of a circle (arcs and angles)

Hi there. You gave me quite a challenge. There are many pieces in this 
puzzle. Here's a labeled diagram:

        

Now you need to think of a strategy to solve the problem. Notice that 
the segments you are looking for, OC and OB, are sides of a triangle. 
Therefore, if you can establish that triangle OCB is a right triangle,  
you can use the Pythagorean theorem to find the lengths of its sides. 

Here are the arguments you will need. 

(1) Since you are told that diameter DE is perpendicular to chord AB, 
    then angle OCB is a right angle. (Angles at the foot of a 
    perpendicular are right angles.) Therefore, by definition, 
    triangle OCB is a right triangle. 

(2) The Pythagorean theorem says that in a right triangle, 
    a2 + b2 = c2, where a and b are the legs, and c is the hypotenuse. 
    In terms of triangle OCB, the legs are OC and BC, and the 
    hypotenuse is OB. Therefore, OC2 + BC2 = OB2. 

(3) You also know that if a line through the center of a circle is 
    perpendicular to a chord, it also bisects the chord. Therefore, 
    DE bisects AB at point C, and since AB = 24, then AC = BC = 12. 
    This gives you a measure of BC, one of the sides of the triangle.

(4) To answer part (a) of your problem, you have to use the theorem 
    that says that if two chords intersect in a circle, the products 
    of the segments of each chord are equal. 

    In this figure, point C is the point of intersection of chords 
    DE and AB. (Remember, diameter DE is also a chord.) Therefore, 
    EC x DC = AC x CB. It was given that DC = 8, and you know that 
    AC and CB = 12. Use the theorem to form the equation and solve 
    for EC: 

    8 * EC = 12 * 12
    8 * EC = 144
        EC = 18.

(5) Since EC = 18 and CD = 8, and EC + CD = ED, then the diameter 
    DE = 26. This means that the radius of this circle is 
    1/2 (26) = 13. Since OB is a radius, OB = 13. 

(6) Now that you have two of the three sides of the triangle, you can 
    substitute the values of the two sides that you know into the 
    Pythagorean theorem and find x, the length of OC.

I've done the hard part and I leave the rest up to you. If you can't 
figure out the last part, let me know.

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/