Chapter Closure
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Segments of Chords Theorem |
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If two chords intersect in a circle, then the products of the lengths of the chord segments are equal. |
Using this theorem, we can write an equation containing the given lengths in the diagram.
Recall the Perpendicular Chord Bisector Theorem.
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Perpendicular Chord Bisector Theorem |
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If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. |
This means AD bisects EC. Therefore, it must be that EF≅ FC. Let's add this to the diagram.
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Inscribed Angles Theorem |
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The measure of an inscribed angle is half the measure of its intercepted arc. |
Using this theorem we can write an equation containing the measures of angles given in the diagram.
LHS * 2=RHS* 2
Distribute 2
LHS-8x=RHS-8x
LHS+5=RHS+5
Rearrange equation
To find b and c we will first consider the Tangent to Circle Theorem.
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Tangent to Circle Theorem |
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In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle. |
With this information we can identify the following right angle in the diagram.
Let's add some more information to the diagram. Notice that an arc and its corresponding central angle are congruent. If we draw a segment from C to our tangent's point of intersection, this segment will bisect the central angle.
Now we can identify two right triangles where one of the non-right angles is 60^(∘). As we can see, there are two 30^(∘)-60^(∘)-90^(∘) triangles. Such a triangle has a longer leg that is sqrt(3) times the length of the shorter leg. Now we can determine the value of c.
We see that c=5sqrt(3). The value of b is the sum of the triangle's 30^(∘) angles. Therefore, it must be that b=60^(∘).