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In the semicircle above, the length of arc AC is equal to the length of arc BD, and the length of arc AB is less than the length of arc BD.

Quantity A

Quantity B

\frac{\text{the length of chord AB}}{\text{the length of chord CD}}

\frac{1}{2}

A) Quantity A is greater. B) Quantity B is greater. C) The two quantities are equal. D) The relationship cannot be determined from the information given.

In the semicircle above, the length of arc AC is equal to the length of arc BD, and the length of arc AB is less than the length of arc BD.

Quantity A

Quantity B

\frac{\text{the length of chord AB}}{\text{the length of chord CD}}

\frac{1}{2}

A) Quantity A is greater. B) Quantity B is greater. C) The two quantities are equal. D) The relationship cannot be determined from the information given.

Okay, so here's what we have so far... As you can see, I've added the entire circle AND I've added the circle's center

Now the question tells us that the length of arc AB is less than the length of arc BD. However, at this point, I want to investigate what would happen if it were the case that the length of arc AB is equal to the length of arc BD We'd get something like this... Notice that all 3 arcs (AC, AB, and BC) all have the SAME length.

This means that each CENTRAL ANGLE that "holds" these 3 equal arcs must also be equal...

Since all three angles are on the same line (the diameter to be exact), they must add to 180°, which means each angle must be 60°

Since OA and OB are the radii of the circle, we can conclude that ∠OAB and ∠OBA must both be equal, which means they both equal 60°

So, ∆OAB is an EQUILATERAL TRIANGLE, which means all 3 sides have equal length. In fact all 3 sides are equal to the radius of the circle.

Since OC and OD are also radii, we can see that we have a BUNCH of line segments that are all the same length.

At this point, we can see that: (length of chord AB)/(length of chord CD) = 1/2 [since AB = the length of 1 radius, and CD = the length of 2 radii]

So, if it were the case that the length of arc AB is equal to the length of arc BD, then Quantities A and B would be EQUAL.

However, the original question tells us that the length of arc AB is less than the length of arc BD. From this, we can conclude that chord AB is LESS THAN the radius of the circle.

This means (length of chord AB)/(length of chord CD) < 1/2