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 $AE.$

$\dfrac{4\sqrt5}{3}$

$\dfrac{5\sqrt5}{3}$

$\dfrac{12\sqrt5}{7}$

$2\sqrt5$

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

Answer: B

Concepts:coordinate geometry distance formula system of equations

Difficulty rating: 1690

Solution:

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.

2000 AMC 10 Problem 16

Answer: B

Difficulty rating: 1690

Solution:

Problem 16 in Other Years