The heights of 4 athletes on a team are $135 \mathrm{~cm}, 160 \mathrm{~cm}, 170 \mathrm{~cm}$, and $175 \mathrm{~cm}$. Laurissa joins the team. On the new team of 5 athletes, the mode height of the players is equal to the median height which is equal to the mean (average) height. How tall is Laurissa?
(A) $135 \mathrm{~cm}$
(B) $160 \mathrm{~cm}$
(C) $170 \mathrm{~cm}$
(D) $175 \mathrm{~cm}$
(E) $148 \mathrm{~cm}$
Since the heights of the 4 athletes on the team are all different, then if Laurissa's height is different than each of these, there is no single mode height.

Therefore, Laurissa's height must be equal to the height of one of the 4 athletes on the team for there to be a single mode.

If Laurissa's height is $135 \mathrm{~cm}$, then the median height of the 5 athletes is $160 \mathrm{~cm}$ which is not possible, since the median does not equal the mode.

Similarly, if Laurissa's height is $175 \mathrm{~cm}$, then the median height of the 5 athletes is $170 \mathrm{~cm}$ which is not possible.

Therefore, Laurissa's height must equal $160 \mathrm{~cm}$ or $170 \mathrm{~cm}$, since in either case the median height of the 5 athletes will equal Laurissa's height, which is the mode.

If Laurissa's height is $170 \mathrm{~cm}$, then the mean height of the 5 athletes is

$\frac{135+160+170+170+175}{5}=162 \mathrm{~cm}$.

If Laurissa's height is $160 \mathrm{~cm}$, then the mean height of the 5 athletes is

$\frac{135+160+160+170+175}{5}=160 \mathrm{~cm}$.

When Laurissa's height is $160 \mathrm{~cm}$, the heights of the 5 athletes (measured in $\mathrm{cm}$ ) are: $135,160,160,170,175$.

In this case, each of the mode, median and mean height of the 5 athletes equals $160 \mathrm{~cm}$.