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,(Λ).
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 d≤2AC​,
d(1/2)l​=(1/2)AC(1/2)BD​, or l=2kd where k=ACBD​= constant.
For d≥2AC​,
AC−d(1/2)l​=(1/2)AC(1/2)BD​, or l=2k(AC−d)=−2kd+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 −2k is negative for d>AC/2. Hence (D) is the correct choice.