Question

Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21.$ Including $\overline{AB}$ and $\overline{BC},$ how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}?$

$5$

$8$

$12$

$13$

$15$

Solution:

Let $P$ be the foot of the altitude from $B$ to $\overline{AC}.$ We also get that $AC = 29.$

This tells us that $\dfrac{29 \cdot PB}{2} = \dfrac{20 \cdot 21}{2}$ $PB = \dfrac{20 \cdot 21}{29},$ by calculating the area in two ways. This value is between $14$ and $15.$

Note that as we move the line segment from $\overline{AB}$ to $\overline{PB},$ the line segment's length ranges from $AB$ to $PB.$

The integer values it covers therefore goes from $20$ to $15.$ Similarly, as the line segment moves from $\overline{PB}$ to $\overline{CB},$ its takes on the values from $15$ to $21.$

This gives us $13$ unique line segments that have an integer value length.

Thus, D is the correct answer.

2018 AMC 10A Problem 16

Solution:

Problem 16 in Other Years