Problem: In a rhombus, ABCD, line segments are drawn within the rhombus, parallel to diagonal BD, and terminated in the sides of the rhombus. A graph is drawn showing the length of a segment as a function of its distance from vertex A. The graph is:
Answer Choices:
A. A straight line passing through the origin.
B. A straight line cutting across the upper right quadrant.
C. Two line segments forming an upright V.
D. Two line segments forming an inverted V,(Ξ).
E. None of these.
Solution:
Let d denote the distance from A measured along AC and let l(d) denote the length of the segment parallel to BD and d units from A. Then by similar triangles, we have:
For dβ€2ACβ,
d(1/2)lβ=(1/2)AC(1/2)BDβ, or l=2kd where k=ACBDβ= constant.
For dβ₯2ACβ,
ACβd(1/2)lβ=(1/2)AC(1/2)BDβ, or l=2k(ACβd)=β2kd+2kAC.
The graph of l as a function of d evidently is linear in d. Its slope 2k is positive for d<AC/2; its slope β2k is negative for d>AC/2. Hence (D) is the correct choice.