Geometry

Analyze Triangle Properties:

 

We are given that triangle ABC is a right triangle with angle A = 30 degrees and angle C = 90 degrees. This makes angle B = 60 degrees (since the angles in a triangle sum to 180 degrees).

 

Since BD bisects angle ABC, we know that angles ABD and CBD are both equal to half of angle ABC, which is 30 degrees.

 

Utilize Isosceles Triangle Property:

 

Because BD bisects angle B and is perpendicular to side AC (since AC is the hypotenuse of right triangle ABC), segment BD also bisects side AC. This makes triangles ABD and BDC congruent.

 

Find Lengths in Triangle ABD:

 

Knowing AC = 12 and that BD bisects AC, we can determine that AD = DC = 12 / 2 = 6.

 

Apply Area Formula for Triangle ABD:

 

The area of a triangle can be calculated using the formula: Area = 1/2 * base * height.

 

In triangle ABD, we know the base (AB) is part of the hypotenuse of right triangle ABC. Since angle A = 30 degrees, we can use the 30-60-90 right triangle properties to find that AB = 2 * AC = 2 * 12 = 24.

 

The height of triangle ABD is the perpendicular distance from point D to side AB. Since BD is perpendicular to AC, the height is the same length as segment BD. However, we haven't yet determined the length of BD.

 

Find BD using Pythagorean Theorem:

 

Since triangle ABD is a right triangle with AB = 24 and AD = 6, we can use the Pythagorean Theorem to find BD (the hypotenuse):

 

BD^2 = AB^2 - AD^2 BD^2 = 24^2 - 6^2 BD^2 = 576 - 36 BD^2 = 540 BD = √540 (Since we want the positive side length)

 

However, working with the square root of 540 might be cumbersome. We can simplify this by noticing that 540 can be factored as 2^2 * 3^3 * 5. Taking the square root of both sides:

 

BD = √(2^2 * 3^3 * 5) = 2 * 3 * √5

 

Calculate Area of Triangle ABD:

 

Now that we know all side lengths, we can find the area of triangle ABD:

 

Area = 1/2 * base * height

 

Area = 1/2 * 24 * (2 * 3 * √5)

 

Area = 72 * √5