Question

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?

Answer 1

Segment AD must bisect ∠ A.

Answer 2

The proportion
`(CA)/(CD)=(BA)/(BD)` holds when AD is the bisector of ∠ A that is ∠BAD = ∠CAD

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
`(CA)/(CD)=(BA)/(BD)`

We have to find what must be true about the segment AD so that the proportion
`(CA)/(CD)=(BA)/(BD)` 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
`(CA)/(CD)=(BA)/(BD)` holds when AD is the bisector of ∠ A.

that is ∠BAD = ∠CAD

Thus, The proportion
`(CA)/(CD)=(BA)/(BD)` holds when AD is the bisector of ∠ A. that is ∠BAD = ∠CAD