Answer
See explanations.
Work Step by Step
Step 1. To prove the areas $A_1, A_2, ..., A_5...$ in the figure are all equal, select the $i$th triangle with area $A_i$
Step 2. Because the object is moving with a constant speed, the base of the triangle equals to $b=v\Delta t$ as shown in the figure of the Exercise.
Step 3. The height of the triangle is the distance from the origin to the line of motion $D$, and this distance is the same of all the triangles with areas $A_1, A_2, ..., A_5 ...$
Step 4. The area for the $i$th triangle is given by $A_i=\frac{1}{2}bD=\frac{1}{2}v\Delta tD$
Step 5. With $\Delta t=1$ and constant $v, D$ for all the triangles, all the areas $A_1, A_2, ..., A_5 ...$ will be equal to $A_i$.
