What is the ratio $\frac{A C}{B C}$ for the following construction:
A line segment AB is drawn.
A single ray is extended from A and 12 arcs of equal lengths are cut, cutting the ray at $A_{1}, A_{2} \dots A_{12}$ .
A line is drawn from $A_{12}$ to B and a line parallel to $A_{12} B$ is drawn, passing through the point $A_{6}$ and cutting AB at C.

1:2

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Solution

The correct option is B
1:1

In the construction process given, triangles $△ A A_{12} B$ and $△ A A_{6} C$ are similar.

Hence, we get $\frac{A C}{A B} = \frac{6}{12} = \frac{1}{2}$ .

By construction $\frac{B C}{A B} = \frac{6}{12} = \frac{1}{2}$ .

$\frac{A C}{B C} = \frac{\frac{A C}{A B}}{\frac{B C}{A B}}$

$= \frac{\frac{1}{2}}{\frac{1}{2}} = 1.$

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The correct option is B 1:1 In the construction process given, triangles △AA12B and △AA6C are similar. Hence, we get ACAB=612=12. By construction BCAB=612=12. ACBC=ACABBCAB =1212=1.