2000 AMC 10 Problem 16

Question

The diagram shows

28

lattice points, each one unit from its nearest neighbors. Segment

AB

meets segment

CD

at

E.

Find the length of segment

AE.

\dfrac{4\sqrt5}{3}

\dfrac{5\sqrt5}{3}

\dfrac{12\sqrt5}{7}

2\sqrt5

\dfrac{5\sqrt{65}}{9}

Assign coordinates

A = (0, 3),

B = (6, 0),

C = (4, 2),

D = (2, 0)

Find the equations of lines

AB

and

CD

and solve for their intersection

E

Place the points at

A = (0, 3),

B = (6, 0),

C = (4, 2),

D = (2, 0).

Line

AB

is

x + 2y = 6

and line

CD

is

x - y = 2.

Solving simultaneously gives

E = \left(\dfrac{10}{3}, \dfrac{4}{3}\right).

Then

\begin{aligned} AE &= \sqrt{\left(\dfrac{10}{3}\right)^2 + \left(\dfrac{4}{3} - 3\right)^2} \\ &= \sqrt{\dfrac{100}{9} + \dfrac{25}{9}} \\ &= \dfrac{5\sqrt5}{3}. \end{aligned}

Thus, the correct answer is B.

Problem 16 in other years