math geometry question?

You are looking for the area of ABED which is the area of an equilateral triangle minus the area of a right triangle.  Let's find the area of the right triangle, first.

If AB = 12 and ABC is an equilateral triangle, then BC is also 12.

Since we are told BE is 8, CE is then 4.  We have the base of the triangle.  Once we have the height we can find the area.

Since ABC is an equilateral triangle, angles A, B, and C are 60°.  We know angle CED is 90°, which leaves CDE to be 30° making this a 30-60-90 triangle.

Knowing this, and the base has 4 units, the height is then √3 times that length or 4√3 units.

Now that we know this, we can find the area:

A = bh/2
A = 4(4√3) / 2
A = (16√3) / 2
A = 8√3 unit²

Now let's find the area of the equilateral triangle.  If we drop a line down from B so its perpendicular to AC, we cut the base (12 units) in half (6 units) and create another 30-60-90 triangle.  Again, the height is √3 times the base, so the height is 6√3 units.

Using the base of 12 units we can find the area of this triangle:

A = bh/2
A = 12(6√3)/2
A = 6(6√3)
A = 36√3 unit²

Finally, subtract the smaller triangle out of the larger triangle to get the remaining area you are looking for:

36√3 - 8√3
(36 - 8)√3
28√3 unit²

Answer B

ΔABC is equilateral.
Hence, BC = AB = 12 cm  and  ∠C = 60°

In right ΔDEC:
tan(60°) = DE/[(12 - 8) cm]
DE = 4 tan(60°) cm
DE = 4√3 cm

Area of quadrilateral ABCD
= Area of ΔABC - Area of ΔDEC
= [(1/2) × 12² × sin(60°) - (1/2) × (12 - 8) × 4√3] cm²
= [(1/2) × 144 × (√3/2) - (1/2) × 4× 4√3] cm²
= 28√3 cm²

The answer: B. 28√3

  In the given figure, 
  line segment DE is perpendicular to side BC of equilateral triangle ABC. 
  If AB = 12 and BE = 8, 
  what is the area of the quadrilateral ABED?
  Answer choice:
 (B) 28√3

CE = 12 - 8 = 4
CD = 2 * CE = 8
ED = sqrt(8^2 - 4^2) = sqrt(48) = 4 * sqrt(3)

(1/2) * CE * ED =>
(1/2) * 4 * 4 * sqrt(3) =>
8 * sqrt(3)

Now find the area of ABC

s = (12 + 12 + 12) / 2 = 36/2 = 18

A^2 = s * (s - 12) * (s - 12) * (s - 12)
A^2 = 18 * 6 * 6 * 6
A^2 = 3 * 6^4
A = 6^2 * sqrt(3)
A = 36 * sqrt(3)
Now subtract the area of CDE from the area of ABC.  That will give you the area of ABED

36 * sqrt(3) - 8 * sqrt(3) =>
28 * sqrt(3)