2001 AMC 10 Problem 15

1. Picture the crosswalk. Its four edges are two pieces of curb (parallel to each other) and two stripes (parallel to each other). A four-sided shape with both pairs of opposite sides parallel is a parallelogram.
2. The distance between the stripes is a height of this parallelogram, measured straight across from one stripe to the other. Since the area of a parallelogram is base times height, find the area another way and then divide.
3. First way. Use a $15$-foot curb piece as the base. The height is the straight-across distance between the curbs, which is $40$ feet. Area $= 15 \times 40 = 600$ square feet.
4. Second way. Use a $50$-foot stripe as the base. The height is the distance $d$ between the stripes. Area $= 50 \times d$.
5. It is the same shape, so the areas are equal: $50d = 600$, and $d = 12$.

The answer is $\boxed{\textbf{(C)}\ 12}$.

A parallelogram has two different base-and-height pairs, and both give the same area. If you know one pair and one side of the other pair, the missing height is just the area divided by that side. "Distance between two parallel lines" always means the straight-across (perpendicular) distance, so it is a height.

Use similar right triangles. A stripe crosses $40$ feet of street and is $50$ feet long. It forms a right triangle with sides $30$, $40$, $50$ (it moves $30$ feet along the street). Now drop a line from the end of the $15$-foot curb piece straight across to the other stripe. This makes a smaller right triangle with the $15$ feet as its longest side, and it has the same shape as the $30$-$40$-$50$ triangle. The distance $d$ matches the $40$ side, so $d = 15 \times \frac{40}{50} = 12$.

The two curbs are parallel and the two stripes are parallel, so the crosswalk is a parallelogram. Its sides along the curbs have length $15$ and its other two sides (the stripes) have length $50$.

Compute its area two ways. With the curb segments as bases, the height is the width of the street:
$$\text{Area} = 15 \cdot 40 = 600.$$
With the stripes as bases, the height is exactly the distance $d$ between the stripes:
$$\text{Area} = 50 \cdot d.$$
Equating, $50d = 600$ and $d = 12$.

The answer is $\boxed{\textbf{(C)}\ 12}$.