Question

In triangle ABC, (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 (tion form lowest terms).

a. (3/4)
b. (4/3)
c. (16/19)
d. (19/16)

Answer 1

To find the ratio of the area of triangle ABD to the area of triangle ACD, we can use the angle bisector theorem and the given lengths of the sides of triangle ABC to find the lengths of AD and CD. Once we have the lengths, we can calculate the areas and determine the ratio.

Step-by-step explanation:

To find the ratio of the area of triangle ABD to the area of triangle ACD, we need to find the lengths of AD and CD, which are the angle bisectors of triangle ABC.

To find AD, we can use the angle bisector theorem. It states that the ratio of the lengths of two angle bisectors in a triangle is equal to the ratio of the lengths of the sides opposite those angles. So, we have (AB/BD = AC/CD). Substituting the given values, we have (16/BD = 24/CD).

Now, we can solve for BD and CD, which will give us the lengths of AD and CD. Once we have those lengths, we can use them to find the areas of triangle ABD and triangle ACD, and then calculate their ratio.