Given Problem:

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. Find the length of BE.

Step 1: Understand the Triangle and Coordinates

Since triangle ABC is equilateral with each side 12 units, we can place it in a coordinate system for easier calculation.

Let’s set:

B at (0, 0)

C at (12, 0)

To find A, we note that in an equilateral triangle, the height from A to BC is $h=\frac{\sqrt{3}}{2}\times \mathrm{side}=\frac{\sqrt{3}}{2}\times 12=6\sqrt{3}$.

The x-coordinate of A is the midpoint of BC, which is at (6, 0). So, A is at $(6,6\sqrt{3})$.

Step 2: Locate Point D (Foot of Perpendicular from A to BC)

BC lies along the x-axis from (0,0) to (12,0). The foot of the perpendicular from A to BC will have the same x-coordinate as A and y=0.

So, D is at $(6,0)$.

Step 3: Locate Point E (Midpoint of AD)

A is at $(6,6\sqrt{3})$, D is at $(6,0)$.

Midpoint E of AD is:

$E=(\frac{6+6}{2},\frac{6\sqrt{3}+0}{2})=(6,3\sqrt{3})$

Step 4: Find Length BE

B is at (0, 0), E is at $(6,3\sqrt{3})$.

Using the distance formula:

$BE=\sqrt{{(6-0)}^2+{(3\sqrt{3}-0)}^2}=\sqrt{6^2+{\left(3\sqrt{3}\right)}^2}$

$=\sqrt{36+9\times 3}=\sqrt{36+27}=\sqrt{63}$

$=\sqrt{9\times 7}=3\sqrt{7}$

Final Answer:

The length of BE is $\sqrt{63}$ units.

Related Topics & Formulae:

Equilateral Triangle Properties: All sides equal, all angles 60°. Altitude = $\frac{\sqrt{3}}{2}\times \mathrm{side}$.

Distance Formula: For points (x1, y1) and (x2, y2), distance = $\sqrt{{(x2-x1)}^2+{(y2-y1)}^2}$.

Midpoint Formula: Midpoint of (x1, y1) and (x2, y2) is $(\frac{x1+x2}{2},\frac{y1+y2}{2})$.

Foot of Perpendicular: From a point to a line (here x-axis), the foot has same x-coordinate and y=0.