Question

In triangle ABC, where AB=16, AC=24, BC=19, and AD is an angle bisector. Find the ratio of the area of triangle ABD to the area of triangle ACD in lowest terms.

a) 16:23
b) 8:11
c) 4:5
d) 2:3

Answer 1

The ratio of the area of triangle ABD to the area of triangle ACD is 2:3, obtained through the Angle Bisector Theorem and the given side lengths.

Step-by-step explanation:

To find the ratio of the areas of triangles ABD to ACD in triangle ABC with AB=16, AC=24, BC=19, and AD being an angle bisector, we can use the Angle Bisector Theorem. This theorem states that the angle bisector in a triangle divides the opposite side into two segments that are proportional to the adjacent sides.

We can say BD/CD = AB/AC using the Angle Bisector Theorem. Now, we plug in the given lengths to get BD/CD = 16/24, which simplifies to 2/3 or BD/CD = 2/3. Since AD is an angle bisector, triangles ABD and ACD share the height, and thus, the ratio of their areas is equal to the ratio of their bases BD and CD.

Therefore, the area ratio of triangle ABD to ACD is also 2/3, which simplifies to 2:3. So, the correct answer is d) 2:3.