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Word Lesson: Circles - Properties of Chords
In order to solve problems which involve intersecting chords in circles, it is necessary to
 
  1. The angle formed by intersecting chords is ½ the sum of the intercepted arcs.
  2. The products of the segments formed by intersecting chords are equal.
 
A typical problem involving the angles and segments formed by intersecting chords in a circle gives us information about the lengths of parts of the chords, about the angles formed by the chords, and/or about the arcs of the circle intercepted by these angles. Two examples of this type of problem follow:
 
  1. In circle O shown below, chords CB and AD intersect at point P. The segments formed by these intersecting chords are CP = 7, BP = x, AP = 2x, and DP = x + 1. What is the measure of chord CB?
 
 
We note that because the chords intersect, we have
 
(CP)(BP) = (AP)(DP)
7x = 2x(x + 1)
7x = 2x2 + 2x
 
To solve for x, we will collect like terms and set our equation equal to zero.
 
2x2 - 5x = 0
x(2x - 5) = 0
x = 0     or    x =5/2
 
Although x = 0 is an answer, this would make BP = 0. We will use x = 5/2.
 
CB = CP + BP
CB = 7 + x
CB = 7 + 5/2
CB = 19/2
 
  1. In circle O given below, suppose that angle 3 is 40° and angle B is 100°. What is the measure of angle 1?
 
 
Notice that the arc CB is intercepted by angle 3.
 
Since angle 3 is 40°, we know that arc CB is 80°.
 
Similarly, we know that arc APD is 200° since angle B is 100°
 
Since angle 1 is formed by intersecting chords, it has a measure equal to one half the sum of the intercepted arcs CB and APD.
 

Examples
Example Trumpet Two chords intersect in a circle. The segments of one chord have lengths 5 and x + 2. The segments of the other chord have lengths x and 2x + 3. What are the lengths of these chords?
What is your answer?
 
Example
Trumpet In circle O given below, AB is a diameter. If arc AC is 30° and angle 1 is 70°, what are the measures of arcs CB, DB, and AD?
 
What is your answer?
 

Examples
Example Trumpet Two chords intersecting in a circle form a 50° angle. The intercepted arcs have measures of 2x and 5x – 6, what are the lengths of the arcs?
  1. the measures are 30.29° and 69.71°
  2. the measures are 61.14° and 26.86°
  3. the measures are 50° and 50°
  4. the measures are 8.86° and 16.14°
What is your answer?
 
Example Trumpet Two intersecting chords are divided into segments. One chord has segments of lengths x – 2 and 20. The other chord has segments of lengths x + 4 and 3. What are the lengths of the chords accurate to two decimal places?
  1. no possible answer because x is negative
  2. 10.06 and 21.06
  3. 14.31 and 25.31
What is your answer?
 

When chords intersect in a circle, we can make conclusions about the angles formed and about the segments into which the chords divide each other. The two key facts are:
 
  1. The angle formed has a measure equal to (1/2) the sum of the intercepted arcs.
  2. The products of the segments formed by the intersecting chords are equal.
 
When using these two basic facts, common errors are to mishandle the factor of (1/2) and to multiply the wrong segments together. Care must be taken to use the (1/2) correctly and to add the appropriate intercepted arcs. Care must also be taken to multiply the two segments of the same chord when setting up an equation involving the lengths of the chords.
M Ransom

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