Ellen has property whose boundary lines form a triangle, as shown in the diagram. her house lies at vertex a, and she is going to put a bunkhouse for the ranch hands somewhere along the segment bc. she wants the placement to be such that ca cd = ba bd . what must be true about the segment ad so that the proportion ca cd = ba bd holds?

The proportion  holds when AD is the bisector of ∠ A that is ∠BAD = ∠CAD

Step-by-step explanation:

Given : Ellen has property whose boundary lines form a triangle. Her house lies at vertex A, and she is going to put a bunkhouse for the ranch hands somewhere along the segment BC.

She wants the placement to be such that  

We have to find what must be true about the segment AD so that the proportion  holds.

CONVERSE OF ANGLE BISECTOR THEOREM  states that if a point D on the side BC of triangle ABC divides BC in the same ratio as the sides AB and AC, then AD is the angle bisector of angle ∠ A.

Thus, applying converse of angle bisector theorem,

The proportion  holds when AD is the bisector of ∠ A.

that is ∠BAD = ∠CAD

Thus, The proportion  holds when AD is the bisector of ∠ A. that is ∠BAD = ∠CAD