Illuminations: Soccer Problem

Soccer Problem
  A soccer player is on a breakaway, dribbling the ball downfield, parallel to a sideline. From where should she shoot to have the best chance of making a goal? That is, at what point will the angle formed by the player and the two goal posts be the greatest?

The applet below allows you to investigate this problem by changing the location of the player as well as the distance between the player and the goal posts. As you move the player, the angle changes. Where should the player be placed so that the angle is maximized?

Instructions
 
Adjust the Distance to Goal slider to change how far player P is from the middle of the goal.

Move the player to change the angle formed by the player P and the two goal posts, A and B. Obviously, the larger the angle, the better the player's chance of making a goal.

The Show/Hide Circle buttons draw a circle tangent to the breakaway line that passes through A and B.

The Trace buttons show the path of point P as it is moved or as the distance to goal is changed.

Exploration
 
Drag point P so that the angle is as large as possible along that breakaway line.
  • Click Show Circle. What is the relationship between point P and the circle?
  • Change the Distance to Goal, and again find the maximum angle. What is the relationship between P and the circle at this new distance?

Find the maximum angle, then click Trace followed by Turn Off Trace. This will place a point at P. Change the Distance to Goal, and repeat. Do this several times at various distances.

  • What is the shape of the curve formed by this collection of points?
  • Place P on the breakaway line so that the angle is maximized. Turn on Trace, and change the Distance to Goal. This will draw a trace of all the points P at which the angle is maximized. What is the shape of this path?

   Sorry, this page requires a Java-compatible web browser.

NCTM Resources

Navigating through Geometry in 9‑12

Lessons
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
© 2000 National Council of Teachers of Mathematics
Use of this Web site constitutes acceptance of the Terms of Use

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.