What is the ratio $\frac{A B}{B C}$ for the line segment AB following the construction method below?
Step 1. A ray is extended from A and 30 arcs of equal lengths are cut, cutting the ray at $A_{1}, A_{2}, . . ., A_{30}$
Step 2. A line is drawn from $A_{30}$ to $B$ and a line parallel to $A_{30} B$ is drawn, passing through the point $A_{17}$ .

17:30

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Solution

The correct option is B
17:13

Here the total number of arcs is equal to m+n in the ratio m:n.

The triangles $△$ $A A_{17} C$ and $△$ $A A_{30} B$ are similar.

Hence, $\frac{A C}{A B}$ = $\frac{A A_{1} 7}{A A_{3} 0}$ = $\frac{17}{30}$

1 = AB/AB = $\frac{A C}{A B}$ + $\frac{B C}{A B}$
$\frac{B C}{A B}$ = 1 - $\frac{17}{30}$ = $\frac{13}{30}$

Hence, $\frac{A B}{B C}$ = $\frac{17}{13}$ = $17 : 13$ .

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What is the ratio $\frac{A C}{B C}$ for the line segment AB following the construction method below?
Step 1. A ray is extended from A and 30 arcs of equal lengths are cut, cutting the ray at $A_{1}, A_{2}, . . ., A_{30}$
Step 2. A line is drawn from $A_{30}$ to $B$ and a line parallel to $A_{30} B$ is drawn, passing through the point $A_{17}$ and meet AB at C.

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What is the ratio ABBC for the line segment AB following the construction method below? Step 1. A ray is extended from A and 30 arcs of equal lengths are cut, cutting the ray at A1,A2,...,A30 Step 2. A line is drawn from A30 to B and a line parallel to A30B is drawn, passing through the point A17.

The correct option is B 17:13 Here the total number of arcs is equal to m+n in the ratio m:n. The triangles △ AA17C and △ AA30B are similar. Hence, ACAB = AA17AA30 = 1730 1 = AB/AB = ACAB + BCAB BCAB = 1 - 1730 = 1330 Hence, ABBC = 1713 = 17:13.