Super ball

A friction-less board has the shape of an equilateral triangle of side length $1$ meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex $A$ towards the side $BC$. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is ?

#Geometry

Easy Math Editor

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Comments

Consider an array of triangles as shown below, with vertices labelled as shown. The red straight line travels from a vertex A to a vertex B, crossing $9$ edges in the process.

Now fold up the red line, reflecting each portion of the line in the triangle edge it crosses, and we obtain the following track of a ball from vertex A to vertex B, which bounces perfectly off 9 walls.

The original red length moves $5\tfrac12$ m horizontally and $\tfrac12\sqrt{3}$ m vertically, and hence is $\sqrt{31}$ m long. Thus the ball's path is $\sqrt{31}$ m long.

Mark Hennings - 1 year, 9 months ago

Amazing solution @Mark Hennings Sir , btw if ball was thrown from a point in the line AB , except the vertices , can then also the ball will land on a vertice after nine times striking the walls? Is it possible ?