Each side of triangle ABC is 12 units. D is the foot of the perpendicular dropped from A on BC, and E is the midpoint of AD. The length of BE, in the same unit, is:

1961 AMC 12 Problem 10

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10.

Each side of triangle

ABC

12

units.

D

is the foot of the perpendicular dropped from

A

BC,

and

E

is the midpoint of

AD.

The length of

BE,

in the same unit, is:

\sqrt{18}

\sqrt{28}

6

\sqrt{63}

\sqrt{98}

In the equilateral triangle, find

BD

and

AD

Use right triangle

BDE

after halving

AD

In the equilateral triangle,

BD=6

and

AD=6\sqrt3.

Hence

DE=3\sqrt3,

and right triangle

BDE

gives

\begin{aligned} BE^2&=6^2+(3\sqrt3)^2\\ &=36+27=63. \end{aligned}

Therefore

BE=\sqrt{63}.

Thus, the correct answer is D.