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 $45$

Answer: D

Concepts:similarity arithmetic sequence summation

Difficulty rating: 1530

Small Hint:

Each small triangle with vertex $A$ is similar to triangle $ABC$

Big Hint:

The seven parallel lengths are $\frac{10}{8},\frac{20}{8},\ldots,\frac{70}{8}$

Solution:

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.

1957 AMC 12 Problem 35

Answer: D

Difficulty rating: 1530

Small Hint:

Big Hint:

Solution:

Problem 35 in Other Years