Use only translation - A covering problem in grometry from “100 covering problems”

Question

Given 100 congruent triangles in the plane. Is it always possible to translate 99 of them, such that they completely cover the 100th?

A translation moves every point by the same amount in a given direction.

Answer 1

Here is a counterexample.

Take an extremely long, thin isosceles triangle $T$, say the one with vertices $P_0 = (-10000,0)$, $P_1 = (10000,1)$, $P_2 = (10000,-1)$. Let $R_\theta$ be the rotation by an angle $\theta$ centered at the origin. Choose $\theta = \pi/200$. Let your 100 triangles be $$R_{k \theta}(T) \quad (k=0,....,99) $$ No matter which two of these triangles that you take, if you translate one of them and intersect it with the other, the resulting intersection will be a convex polygon (having at most 4 sides) which has diameter much smaller than $10000/100$. It's not possible for a connected shape of diameter $\ge 10000$ to be covered by $100$ shapes whose diameters are $<10000/100$.

Answer 2

No. imagine skinny triangles that look like line segments unless you zoom in to a high magnification. Fix $100$ different angles in $(0,\pi/2)$, and consider the triangles obtained by rotating one through each of those angles. Given the angles, if the triangles are skinny enough then the area of the intersection of any one with any translate of any other has area less than $1/1000$ times the area of one of the triangles.