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 segment BC. She wants the placement to be such that CA/CD=B/A. What must be true about segment AD so that the proportion CA/CD=BA/BD holds

Answer 1

D) Segment AD must bisect ∠A

Explanation:

Thank you, got it correct on the test

Answer 2

Given that in a ΔABC, Ellen house lies at vertex A and bunkhouse is put somewhere along BC, let it be D.

Please refer the diagram attached.

We are also given that D must be placed such that
`(CA)/(CD)= (BA)/(BD)` holds true.

Let us re write the above equation such that CD and BD lie on one side.

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

Which is nothing but angle bisector theorem.

Hence AD must bisect the angle A then only the given condition will hold true.