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?

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 [tex]\frac{CA}{CD}=\frac{BA}{BD}[/tex]

We have to find what must be true about the segment AD so that the proportion [tex]\frac{CA}{CD}=\frac{BA}{BD}[/tex] 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 [tex]\frac{CA}{CD}=\frac{BA}{BD}[/tex] holds when AD is the bisector of ∠ A.

that is ∠BAD = ∠CAD

Thus, The proportion [tex]\frac{CA}{CD}=\frac{BA}{BD}[/tex] holds when AD is the bisector of ∠ A. that is ∠BAD = ∠CAD