2000 AMC 10 Problem 16

Problem 16 of 25IntermediateAlgebra Geometry

The diagram shows

28

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

AB

meets segment

CD

E.

Find the length of segment

AE.

Show solution

Solution

Place the points at

A = (0, 3),

B = (6, 0),

C = (4, 2),

D = (2, 0).

Line

AB

x + 2y = 6

and line

CD

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.