1957 AMC 12 Problem 35

Question

Side

\overline{AC}

of right triangle

ABC

is divided into

8

equal parts. Seven line segments parallel to

\overline{BC}

are drawn to

\overline{AB}

from the points of division. If

BC=10,

then the sum of the lengths of the seven line segments:

cannot be found from the given information

is

33

is

34

is

35

is

45

Each small triangle with vertex

A

is similar to triangle

ABC

The seven parallel lengths are

\frac{10}{8},\frac{20}{8},\ldots,\frac{70}{8}

By similarity, the segment at the

k

th division point has length

\frac{10k}{8}

for

k=1,\ldots,7.

Their sum is

\begin{aligned} \frac{10}{8}(1+2+\cdots+7) &=\frac{10}{8}\cdot28\\ &=35. \end{aligned}

Therefore, the correct answer is D.

Problem 35 in other years