If a person kicks a ball toward the soccer goal, they have an angle in which to score. Where else on the soccer field can somebody shoot and have the same angle? Referring to the picture above, all three players have the same angle in which to kick toward the goal. The question of this assignment is: if one person is positioned to shoot at the goal, where can other players stand to have the same angle as the first person? We will see that the locations on the soccer field for which this happens forms an arc of a circle; this arc is called el Arco Capaz. The arc depends on a given line segment AB and an angle CAB. In this assignment we will see how to find el arco capaz and why the construction given does indeed work.
For example, if we have the line segment AB and the arc CAB below, the arc drawn is the associated arco capaz.
Finding el Arco Capaz: Start with a segment AB and an angle CAB. Draw the line through A perpendicular to AC. Next, draw the perpendicular bisector to AB. Obtain the point O of intersection on these two lines. Draw the circle centered at O passing through A. The (top) arc between A and B on the circle is the arco capaz. That is, if E is any point on the arc of the circle between A and B (on the opposite side of AB from C), then the angle measure of AEB is equal to the measure of CAB.
You can check out the construction in the interactive web page Arco Capaz webpage.
Problem 1. Suppose we mark three points A, B, and E on a circle. Demonstrate why angle AOB is twice as large as angle AEB. As a consequence, if we move E anywhere on the arc in between A and B, we obtain the same angle measure, since moving E does not change A, O, or B. (HINTS:)
Problem 2. Explain why, in the finished diagram of el arco capaz above, angle AOB is twice angle CAB. The following facts should prove useful: the sum of the angles of each of the two triangles inside the blue quadrilateral have 180 degrees. Also, the top triangle has two equal sides, these sides being the radius of the circle. The corresponding angles are then equal. Finally, the construction was done so that the left angle of the quadratilateral is 90 degrees. This angle consists of angle CAB together with another piece.
Problem 3. Explain why the previous two problems say that the construction described above does work to produce el arco capaz.
Problem 4. The situation of the soccer field is a little different. Find the arc that contains C and for which any point E on the arc has angle AEB equal to angle ACB. Think about what Problem 1 says, and recall a construction from a past assignment.
