10.8k views
3 votes
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?

2 Answers

4 votes
Segment AD must bisect ∠ A.
User Tarun Deep Attri
by
7.9k points
0 votes

Answer:

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

Ellen has property whose boundary lines form a triangle, as shown in the diagram. Her-example-1
User Andy Fusniak
by
7.2k points